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your truth table is shown below:




(p v q) is the or statement
(p v q) -> ~p is the implies statement

the or statement is only false if both p and q are false.

the implies statement is only false if (p v q) is true and ~p is false.

v means or
-> means implies
~ means not

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your truth table is shown below:



v is the or symbol.
~ is the not symbol.
-> is the implies symbol.

the or statement is ~p v q
the implies statement is ~q -> (~p v q)

the or statement is only false if both ~p and q are false.
the implies statement is only false if ~q is true and (~p v q) is false.

there are 4 possible conditions in the truth table because there are 2 variables involved. those variables are p and q. the possible conditions are 2 * 2 (2 possible conditions for each variable used).

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if it starts at 2:16 pm and ends at 2:47 pm, then the duration of the class is 31 minutes because 2:47 minus 2:16 = 31.

there are 60 minutes in an hour.

the minute hand goes through a rotation of 360 degrees in one hour.

the number of degrees the minute hand traverses each minute is equal to 360 / 60 = 6 degrees.

31 minutes * 6 degrees per minute = 186 degrees.

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your equation is y = x^2/2
your graph would look like this:

you set up a table with values of x and corresponding values of y.
your table would look like this:

x	y = x^2/2
-10	50
-9	40.5
-8	32
-7	24.5
-6	18
-5	12.5
-4	8
-3	4.5
-2	2
-1	0.5
0	0
1	0.5
2	2
3	4.5
4	8
5	12.5
6	18
7	24.5
8	32
9	40.5
10	50

if you knew the shape of the curve, you might have been able to get by with plotting less values of x and then finding the corresponding value of y by using the equation.

example:
equation is y = x^2/2
when x is equal to 5, y is equal to 5^2/2 = 25/2 = 12.5

since y = x^2 is a form of the quadratic equation, then you could have determined the general shape just from that fact alone.

you do need to be familiar with what the graph of a quadratic equation looks like in order to be able to minimize the number of points you need to sketch the graph.

if in doubt, start with x = 0 and work up and down in increments of 1 to see if a shape develops.

sometimes you need more values of x to see if a shape develops.

in this case, x from -5 to + 5 would probably have been enough.

jumping to x = -10 and x = +10 would have been enough to show that it's was following the same trajectory as the values of x got bigger or smaller.

the minimum point is at x = 0.

you could see a change in direction when you plotted values of x = 0, then x = -1, then x = 1.

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it should be:



that would come out to be: (2800 + 2050) / 50 = 3850 / 50 = 77

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i got something different.
the denominator was 13 so it looks like you did that right.
the numerator, however, is (-9i - 6) if i did it right.
my scratch paper is shown below:


let me know if you agree or not.

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he attended 6136 theatrical shows over y years.
the average number of shows he attended per year would be equal to 6136 / y.

example:

if it was 20 years, then he attended an average of 6136 / 20 = 306.8 theatrical shows per year.

multiply 306.8 by 20 and you get 6136.

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i)if the dog wags its tail, then the dog is not calm.
ii)either the dog does not wag its tail or the dog is not calm
iii) if the dog is not calm, then the dog wags its tail.

p = the dog wags its tail.
q = the dog is not calm.

~p = the dog does not wag its tail.
~q = the dog is calm.

statement 1 translates to p -> q
statement 2 translates to ~p v q
statement 3 translates to q -> p

statement 1 and 3 are not equivalent
we need to test with statement 2 to see if it's equivalent to either one of the other 2.

when in doubt, you need to do a truth table.
it tells you whether the statement are equivalent or not.

the truth table for these 3 statements is shown in the following picture.

in this table:

statement 1 becomes statement A.
statement 2 becomes statement B.
statement 3 becomes statement C.



The truth table says that (p -> q) is equivalent to (~q v p).

That means that statement 1 is equivalent to statement 2.

It would be difficult to see this without use of the truth table.

Logically, trying to make sense of it takes a little stretch.

I believe the answer lies in the law of disjunction.

The law of disjunction states:

if A or B is true, then:
If A is false, then B must be true.
If B is false, than A must be true.

we have 2 statements that we think are equivalent because the truth table test of equivalency says that they are.

The 2 statements are:

i)if the dog wags its tail, then the dog is not calm.
ii)either the dog does not wag its tail or the dog is not calm

the contra-positive of the first statement and the first statement are equivalent.

The first statement is:
if the dog wags its tail, then the dog is not calm.
The contra-positive to the first statement is:
if the dog is calm, then the dog does not wag its tail.

The second statement is:
either the dog does not wag its tail or the dog is not calm.

If we assume that the dog does not wag its tail is false, then the law of disjunction says that the dog is not calm must be true.

we can write this another way to say:
if the dog wags its tail, then the dog is not calm.
that looks very much like the first statement.

if we assume that the dog is not calm is false, then the dog does not wag its tail must be true by the same law of disjunction.

we can write this another way to say:
if the dog is calm, then the dog does not wag its tail.
that looks very much like the contra-positive to the first statement.

i'm satisfied that the truth table is telling the truth and that statements 1 and 2 are equivalent even though anybody would be hard pressed to determine that they are without using the rules of logic to confirm that.

the truth table test is the way to determine the equivalency of two statements.

that's called demorgan's law i believe.

here's some additional reading for you if i didn't send it to you earlier.
there are references in there as well so you can study to your heart's content if you so desire.

STATEMENT LAWS AND FALLACIES

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if this is supposed to be a straight line, then m is the slope.
the equation of a straight line is equal to:
y = mx + b
m is the slope
b is the y intercept
the slope is given by the equation.
m = (y2-y1)/x2-x1)
the coordinates you are given can be labeled as:
(x1,y1) = (-3,2k)
(x2,y2) = (k,6)
y2 - y1 would be equal to 6 - 2k
x2 - x1 would be equal to k - (-3) which is equal to k + 3
the slope is therefore equal to (6-2k) / (k+3) and is equal to 4.
you need to solve that equation for k.

you have (6-2k) / (k+3) = 4
multiply both sides of this equation by (k+3) to get:
(6-2k) = 4 * (k+3)
simplify to get:
6 - 2k = 4k + 12
subtract 4k from both sides of the equation and subtract 6 from both sides of the equation to get:
-2k - 4k = 12 - 6
simplify to get:
-6k = 6
divide both sides of the equation by -6 to get:
k = -1

when k = -6, your coordinates becomes:
(-3,-2) and (-1,6) with m = 4
(x1,y1) = (-3,-2)
(x2,y1) = (-1,6)
y2 - y1 = 6 - (-2) = 8
x2 - x1 = -1 - (-3) = 2
8/2 = a slope of 4 which confirms the calculations are correct.

your answer is that k = -1

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if i sing in the shower, then i will not be overheard while singing.
i was overheard while singing
therefore i did not sing in the shower.

this looks like it is valid by the law of contra-position.

the law of contra-position states:

if p then q
not q
therefore not p

let p = i sing in the shower
let q = i will not be overheard while singing.

the statement:
if i sing in the shower then i will not be overheads while singing
translates to:
p -> q

i was overhead while singing translates to:
~q

therefore i did not sing in the shower translates to:
~p

the whole argument of:
if i sing in the shower, then i will not be overheard while singing.
i was overheard while singing
therefore i did not sing in the shower.

translates to:
p -> q
~q
therefore ~p

This is valid by the law of contra-position.

here's some additional information that might be helpful.

v means or
^ means and
~ means not
-> means implies
<-> means is equivalent to

STATEMENT LAWS AND FALLACIES

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range is the difference between the lowest and the highest.
the number are:
76,91,95,95,99
put them in order to get:
76, 91, 95, 95, 99
the range would be 99 minus 76 = 23
variance is the sum of the squared differences from the mean divided by the number of entries.
the mean is the average.
76 + 91 + 95 + 95 + 99 = 456 / 5 = 91.2
the variance is calculated as shown in the following table

c1        c2       c3
-------------------------
76	-15.2	231.04
91	-0.2	0.04
95	3.8	14.44
95	3.8	14.44
99	7.8	60.84
--------------------------		
456		320.8      total
91.2		64.16      average (total divided by 5)

the first column is the data
the sum of the data is 456
the mean (average) of the data is 91.2
the second column is the difference between each data item and the mean.
the fact that the number is negative doesn't matter because it will be squared and become positive.
the third column is the square of the difference.
the sum of the squared difference is 320.8
the mean (average) of the squared difference is 64.16
64.16 is your variance
91.2 is your mean
23 is your range

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5 more than the product of a number and 3 would be equal to
(3 * number) + 5
if x is the number, then:
3x + 5

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400 people
student costs $2.00 apiece
Adult costs $3.000 apiece.
total collected was $1050.
x = number of student tickets
y = number of adult tickets.
x + y = 400
2x + 3y = 1050
solve simultaneously.
multiply first equation by 2 to get:
2x + 2y = 800
2x + 3y = 1050
subtract first equation from second equation to get:
y = 250
since x + y = 400, then x = 150.
2*150 + 3*250 = 300 + 750 = 1050.
number of child tickets is 150
number of adult tickets is 250

integers are x and x+1
sum of first and 3 times second is 55.
x + 3(x+1) = 55
x + 3x + 3 = 55
4x + 3 = 55
4x = 52
x = 13
x+1 = 14
13 + 3 * 14 = 13 + 42 = 55.

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interest charge was $234.
6.5% is the annual interest rate percent i presume.
the rate is .065 per year.
the monthly rate is .0054166667 per month.
the total interest is 8 * .0054166667 = .043333333
since this is simple interest, there is no compounding that has to be performed.
divide the interest by the interest rate and you get the principal of $5400.
take 5400 and multiply it by .065 to get $351.
multiply that by 8 and divide it by 12 to get $234.
the money that was borrowed is $5400.

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5 * 100 = 500
12 * 10 = 120
3 * 1 = 3
500 + 120 + 3 = 623

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i'll work in thousands.
you start at 350 thousand in year 0, drop to 300 thousand in year 2 and drop to 250 thousand in year 4.
you are dropping 50 thousand every 2 years or 25000 a year.
your equation is y = -25x + 350
your graph looks like this.

the horizontal lines are at y = 350, 300, and 250 thousand.
when x = 0, y = 350 thousand
when x = 2, y = 300 thousand
when x = 4, y = 250 thousand
the problem statement says:
original projections predicated the same population in half the number of years.
this means that Rochester was projected to have 300 thousand in 1 year and 250 thousand in 2 years.
that would be a drop in population of 50 thousand per year.
the equation for that would be y = -50x + 350
add that equation to the graph and it looks like what you see below.

you can see that the first equation crosses the 300 thousand line in 2 years and the 250 thousand line in 4 years, while the second equation crosses the 300 thousand line in 1 year and the 250 thousand line in 2 years.
answers to your questions are below:
At what rate was the population changing?
it changed at the rate of minus 25 thousand per year.
At what rate did the projections predict the population would change?
the projections predicted the population would change at the rate of minus 50 thousand per year.
What was the predicted population in 2003?
it depends on when you started.
the predicted population decreased by 50 thousand per year.
it started at 350 thousand.
if the start year is 2000, then 2003 is 3 years from that and the predicted population would be 350 thousand minus 3*50 thousand = 350 thousand minus 150 thousand = 200 thousand.
find the start year and subtract it from 2003. multiply 50 thousand by the number of years and subtract it from 350 thousand.

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the roots on this are imaginary.
here's what the graph looks like.

It doesn't cross the x-axis so it has no real roots.
you can solve it by completing the square but it will have a negative square root.
the equation is 4x^2 - 3x + 1
set it equal to 0 and subtract 1 from both sides of the equation to get:
4x^2 - 3x = -1
divide both sides of the equation by 4 to get:
x^2 - .75x = -.25
take half the .75 to get .375
square half the .75 to get .375^2 = .140625 and then add it to the right side of the equation to get.
(x - .375)^2 = -.25 + .140625
combine like terms to get:
(x - .375)^2 = -.109375
take the square root of both sides of the equation to get:
x - .375 = +/- sqrt(-.109375)
add .375 to both sides of the equation to get:
x = .375 +/- sqrt (-.109375)
Since you cannot take the square root of -.109375 and get a real number, the equation has no roots.
this is confirmed by the graph because, if the equation had real roots, the graph of the equation would have crossed the x-axis.
if you need a tutorial on how to find the roots of an equation by completing the square, check out the following reference.
http://www.purplemath.com/modules/sqrquad.htm

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you can convert to kilometers and hours ahead of time, or you can solve in minutes and meters first and then convert to kilometers and hours.
we'll solve for minutes and meters first and then convert.
the basic formula used if rate * time = distance.
you are given time in minutes and distance in meters.
we set the rate of the boat as x.
we set the rate of the current as y.
going against the current, rate * time = distance becomes:
(x-y) * 10 = 1500
without any current, rate * time = distance becomes:
x * 7.5 = 1500
you have 2 equations that need to be solved simultaneously.
use the second equation to replace x in the first equation.
the second equation is 7.5*x = 1500
divide both sides of this equation by 7.5 to get x = 200
substitute for x in the first equation.
the first equation is (x-y) * 10 = 1500
substituting for x gets:
(200-y) * 10 = 1500
divide both sides of this equation by 10 to get:
200 - y = 150
subtract 200 from both sides of this equation to get:
-y = -50
multiply both sides of this equation by -1 to get:
y = 50
you have:
x = 200
y = 50
x is the speed of the boat
y is the speed of the current
the terms rate and speed are used inter-changeably. in this problem they mean the same thing.
substitute for and y in the original equations to see if these values are good.
7.5 * x = 1500 becomes 7.5 * 200 = 1500 becomes 1500 = 1500 which is true, confirming the value of x is good in the first equation.
(x-y) * 10 = 1500 becomes (200-50) * 10 = 1500 becomes 150 * 10 = 1500 becomes 1500 = 1500 which is true, confirming the value of x and y are both good in the second equation.
those are your answer.
they are, however in minutes and meters.
x = 200 means 200 meters per minute.
y = 50 means 50 meters per minute.

to translate meters to kilometers, you need to divide your meters by 1000.
your answer becomes:
x = .2 kilometers per minute
y = .05 kilometers per minute

to translate per minute to per hour, you need to multiply your minutes by 60.
your answer becomes:
x = 12 kilometers per hour.
y = 3 kilometers per hour.

12 kilometers per hour is the same as 12000 meters every 60 minutes.
divide it by 60 and you get 200 meters per minute.

3 kilometers per hour is the same as 3000 meters every 60 seconds.
divide it by 60 and you get 50 meters per minute.

the answers are equivalent.

you can also solve this problem by converting everything to kilometers and hours first and then solving.

if you did that, you would do the following.

1500 meters / 1000 = 1.5 kilometers
10 minutes / 60 = .167 hours
7.5 minutes / 60 = .125 hours

in your calculations use unrounded numbers rather than rounded numbers. store them in memory in your calculator and then recall them when you want to use them.

the two equations we have to work with were originally:
(x-y) * 10 = 1500
7.5 * x = 1500
those equations now becomes:
(x-y) * .167 = 1.5
.125 * x = 1.5
solve for x in the second equation to get:
x = 1.5 /.125 = 12 kilometers per hour.
use that value of x in the first equation to solve for y.
(x-y) * .167 = 1.5 becomes:
(12 - y) * .167 = 1.5
divide both sides of this equation by .167 to get:
12 - y = 9
subtract 12 from both sides of this equation to get:
-y = -3
multiply both sides of this equation by -1 to get:
y = 3
your answer are:
x = 12 kilometers per hour
y = 3 kilometers per hour

you got the same answer whether you used meters and minutes or you used kilometers and hours.

that confirms your answers are good.

the speed of the boat is 12 kilometers per hour.
the speed of the current is 3 kilometers per hour.

the time it takes without the current is 7.5 minutes which equals (7.5)/60 hours.

(7.5)/60 * 12 kilometers per hour = 1.5 kilometers which is equivalent to 1500 meters.

the time it take against the current is 10 minutes which equals (10/60) hours.

(10/60) * (12-3) = (10/60) * 9 = 90/60 = 1.5 kilometers which is equivalent to 1500.

everything checks out so you can be confident that the answer is good.

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if you were more specific about what you wanted to do, it would be easier to answer you.

are there 3 variables involved?

are there 2 variables plus a statement such as OR or AND or IMPLIES or IS EQUIVALENT TO.

give me a couple of more specifics and i might be able to help you.

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you need to divide 68 by (3 and 1/3)
3 and 1/3 is equivalent to 10/3
you need to divide 68 by 10/3.
this is equivalent to multiplying 68 by 3/10.
your answer becomes (68*3)/10 which is equal to 204/10 which is equal to 20.4
the integer portion of 20.4 is the number of (3 and 1/3) feet pieces of wire you can cut from the 68 foot long wire.
the fractional portion is what's left over.
the integer part comes out to be a total of 20 * (3 and 1/3) = (60 + 20/3) which is equal to (66 + 2/3) feet.
the fractional part comes out to be a total of .4 * (3 and 1/3) = 4/10 * (3 and 1/3) = (4/10 * 3) + (4/10 * 1/3) = (12/10 + 4/30)
you have the 20 * (3 and 1/3) feet length that come up to a total of (66 and 2/3 feet).
you have the .4 * (3 and 1/3) feet length that comes up to a total of (12/10 + 4/30 feet). this is equivalent to (36/30 + 4/30) = 40/30 = 4/3 = (1 and 1/3).
add (66 and 2/3) to (1 and 1/3) and you get (67 and 3/3) which comes up to be 68.
the calculations are correct.
your answer is that you can get 20 pieces of (3 and 1/3) foot lengths of wire from the 68 foot length wire with one remaining length of (1 and 1/3) feet.

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company A charges $90.00 per month flat fee.
company B charges $15.00 pr month plus $.10 per minute.
equation for company A is:
y = 90
equation for company B is:
y = .10*x + 15
you can graph both equations.
looks like this:

the plan with the flat rate is more expensive up to about x minutes.
you can solve for the exact value by solving the 2 equations simultaneously.

the 2 equations are:
y = 90
y = .10x + 15
since they both equal to y, then set them equal to each other to get:
90 = .10x + 15
subtract 15 from both sides of the equation to get:
75 = .10x
divide both sides of the equation by .10 to get:
x = 750
the break even point is at 750 minutes.
the graph has been modified to provide a horizontal line at y = 90 and a vertical line at x = 750.
the intersection of the horizontal line and the vertical line should be the break even point.

the horizontal line at y = 90 is the equation for Company A.
The slanted line is the equation for Company B.
It intercepts the y axis at x = 15 which is the cost to the customer with 0 minutes used.
The vertical line shows you that the intersection of the graph of the equation for Company A and the graph of the equation for Company B occurs at x = 750 which is the break even point.
the value of x in the graph is the number of minutes used.
the value of y in the graph is the cost for the number of minutes used.
you can see that the cost for company A customer is $90.00 regardless of the number of minutes they used, while the cost for company B customer has a fixed component ($15.00) and a variable component (10 cents a minute).

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p = probability that the target will be hit = 1/3
q = probability that the target will not be hit = 1 - 1/3 = 2/3
p + q = 1, as it should, since either the target will be hit or the target will not be hit.

the probability that the target will be hit is equal to 1 - the probability that the target will not be hit.

the probability that the target will not be hit is 2/3 * 2/3 * 2/3 because you are firing at the target 3 times and each time the probability that the target will not be hit is 2/3.

the probability that the target will not be hit is therefore 8/27 (2*2*2)/(3*3*3).

since the total probability is 1 which is equivalent to 27/27, then the probability that the target will be hit is equal to 27/27 - 8/27 = 19/27

this is the short way to solve this problem.

the long way is to look at the probability that:

a. the target is hit 1 time only.
b. the target is hit 2 times only.
c. the target is hit 3 times only.

since the target can't be hit any other way, then the probability that the target will be hit will be the sum of these 3 independent events.

the probability that any of the posssible events is given by the formula:

nCx * p^x * q^(n-x)

n is the total number of shots fired at the target.
x is the number of times the target can be hit exactly.

this comes out to be:
3C0 * p^0 * q^3
3C1 * p^1 * q^2
3C2 * p^2 * q^1
3C3 * p^3 * q^0

We already solved for 3C0 * p^0 * q^3
that number came out to be 8/27
3C0 equals 3!/(0!*3!) which equals 1
p^0 = 1
q^3 = (2/3) * (2/3) * (2/3) which equals 8/27
3C0 * p^0 * q^3 = 1 * 1 * 8/27 = 8/27

We can solve for 3C3 * p^3 * q^0 easy enough.
3C3 equals (3! / (3!*0!) which equals 1
p^3 = (1/3) * (1/3) * (1/3) which equals 1/27
q^0 = 1
3C3 * p^3 * q^0 = 1 * 1/27 * 1 = 1/27

We can solve for 3C1 * p^1 * q^2 as follows:
3C1 = 3! / (1!*2!) = 3!/2 = 6/2 = 3
p^1 = 1/3
q^2 = 2/3 * 2/3 = 4/9
3C1 * p^1 * q^2 = 3 * 1/3 * 4/9 = 3/3 * 4/9 = 12/27

We can solve for 3C2 * p^2 * q^1 as follows:
3C2 = 3! / (2!*1!) = 3!/2 = 6/2 = 3
p^2 = 1/3 * 1/3 = 1/9
q^1 = 2/3
3C2 * p^2 * q^1 = 3 * 1/9 * 2/3 = 3/9 * 2/3 = 6/27

our total probabilities are:
p(0) = 8/27
p(1) = 12/27
p(2) = 6/27
p(3) = 1/27

The total probability is 8 + 12 + 6 + 1 = 27/27 = 1 as it should.

The probability we will hit the target is p(1) + p(2) + p(3) = 12 + 6 + 1 = 19/27.

That's the same number we got before when we did 1 - p(0).

We're good and that's your answer.

The probability that the target will be hit is 19/27.

nCx is the combination formula and it is defined as n! / (x! * (n-x)!)

You need to multiply the probability of each occurrence by the number of ways that occurrence can occur.

An example:

You have 3 shots.

You want to get 1 out of 3 shots.

The probability is 3C1 * p(1) * q(2).

Assume h means hit and x means miss.

The number of ways is 3C2 which is equal to 3 (we did that above).

Those possible ways are:

hmm
mhm
mmh

h = hit
m = miss

The probability of each of these ways is the same.

The probability of getting 1 out of 3 is p(hmm) + (pmhm) + p(mmh).

Since the probability of hmm is equal to the probability of mhm is equal to the probability of mmh, then your probability of getting 1 out of 3 can be expressed as 3 * p(hmm).

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(5,1) is the coordinate pair of (x,y)
x is equal to 5 and y is equal to 1.
the equation is 2x - 3y = 7
if you replace x with 5 and y with 1, then the equation should be true.
let's do that.
2x - 3y = 7 becomes:
(2*5) - (3*1) = 7 which becomes:
10 - 3 = 7 which becomes:
7 = 7
the equation is true when we replace x with 5 and y with 1.
that means that the coordinate pair of (5,1) is a solution to the equation of 2x - 3y = 7.

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Identify which argument is invalid.

1. Either the panda yawns or she is alert.
The panda did not yawn.
Therefore, she is alert.

THIS IS VALID BY THE LAW OF DISJUNCTION

2. If the panda yawns, then she is not alert.
The panda is not alert.
Therefore, she yawned.

THIS IS INVALID BY THE FALLACY OF THE CONVERSE.

3. If Tom is cooking, then I am not hungry.
I am hungry.
Therefore, Tom is not cooking.

THIS IS VALID BY THE LAW OF CONTRA-POSITION.

4. If it is hailing, then I will not go outdoors.
If I will not go outdoors, I will not raise any money for charity.
Therefore, if it is hailing, then I will not raise any money for charity

THIS IS VALID BY THE LAW OF SYLLOGISM.

5. If it is hailing, then I am not going outdoors.
It is hailing.
Therefore, I am not going outdoors.

THIS IS VALID BY THE LAW OF DETACHMENT.

This is how the laws work.

LAW OF DETACHMENT

p -> q
p
therefore q

The statement p -> q means "If p is true, then q is true"
As long as p is true, then the statement p -> q is true only if q is true.
If p is true and q is false, then the statement p -> q is false.

Note that the statement p -> q can also be true if p is false, but that's another story we won't get into right now. If p is false, the statement p -> q can be true but you can't tell anything about q from it.

Your statement number 5 fits this.

That statement is:

If it is hailing, then I am not going outdoors.
It is hailing.
Therefore, I am not going outdoors.

We let:
p = It is hailing
q = I am not going outdoors.

your statement translates to:

p -> q
p
therefore q

p -> q translates to "if it is hailing then i am not going outdoors"
p translates to "it is hailing"
therefore q translates to "therefore i am not going outdoors"

This is valid by the law of detachment.

LAW OF SYLLOGISM

p -> q
q -> r
therefore p -> r


The statement p -> q is assumed to be true.
The statement q -> r is assumed to be true.
If both statements are true, then the statement p -> r has to be true.

Your statement number 4 fits this.

That statement is:

If it is hailing, then I will not go outdoors.
If I will not go outdoors, I will not raise any money for charity.
Therefore, if it is hailing, then I will not raise any money for charity



We let:
p = It is hailing .
q = I am not going outdoors.
r = I will not raise any money for charity.

The law of syllogism states:

p -> q
q -> r
therefore p -> r

This translates to:

if it is hailing than I will not go outdoors.
If i don't go outdoors then I will not raise any money for charity.
Therefore if it is hailing, I will not raise any money for charity.

Since the first 2 statements are assumed to be true, then the third statement is assumed to also be true.

LAW OF DISJUNCTION

A or B implies that either A is true or B is true. This means that both cannot be false, otherwise the statement would be false.

If you are given the A or B is true, and you are given that B is false, then you can conclude that A is true. Alternatively, if you are given that A is false, then you can conclude that B is true.

This fits your statement number 1.

That statement is:

Either the panda yawns or she is alert.
The panda did not yawn.
Therefore, she is alert.

Let A = the panda yawns and let B = the panda is alert.
A or B means the panda yawns or the panda is alert.

At least one of those conditions has to be true in order for the OR statement to be true.

Since A is false (the panda did not yawn), then B must be true (the panda is alert.

LAW OF CONTRA-POSITION

The law of contra-position states that:

If a statement is true, then the contra-position of that statement is also true.
Conversely:
If a statement is false, then the contra-position of that statement is also false.

In other words, the statement and its contra-position are equivalent.

In symbolic form, this would show up as:

p -> q is equivalent to ~q -> ~p

p and q are statements.
~p and ~q are negations of those statements.

This can be proven using truth tables but we won't do that here. We will, however, apply the law to your statement number 3.

That statement is:

If Tom is cooking, then I am not hungry.
I am hungry.
Therefore, Tom is not cooking.

Let p = Tom is cooking.
Let q = I am not hungry
let ~q = I am hungry (negation of I am not hungry)
let ~p = Tom is not cooking (negation of Tom is cooking).

The statement is:
p -> q
The contra-position to this statement is:
~q -> ~p

Translating, this comes out to be:

The statement is:
If Tom is cooking, then I am not hungry.
The contra-positive to that statement is:
if I am not hungry, then Tom is not cooking.

Statement 3 fits this so it is valid by the law of contra-position.

FALLACY OF THE CONVERSE

Fallacy of the converse means that A implies B is true does not mean that B implies A is true. B implies A might be true, but B implies A might also not be true (be false).
The two statements are not equivalent.

When you make a statement, there are 3 additional statements that can be made from that.

You have:

the statement
the inverse of the statement
the converse of the statement
the contra-positive of the statement.

In symbolic form, this would be shown as:

the statement:
p -> q
the inverse of the statement:
~p -> ~q
the converse of the statement:
q -> p
the contra-positive of the statement:
~q -> ~p

The rules are:

The statement and the contra-positive of the statement are equivalent.
The inverse of the statement and the converse of the statement are equivalent.

The converse of the statement is not equivalent to the statement.
The inverse of the statement is not equivalent to the statement.

If the statement is true, then:
The converse may or may not be true.
The inverse may or may not be true.
The contra-positive will always be true.

If the statement is false, then:
The converse may or may not be false.
The inverse may or may not be false.
The contra-positive will always be false.

The fallacy of the converse is the assumption that just because the statement is true, then the converse must also be true.

This is not so.

Your statement number 2 fits this form.

That statement is:

If the panda yawns, then she is not alert.
The panda is not alert.
Therefore, she yawned.

Let p = the panda yawns
Let q = The panda is not alert.

The statement becomes:
p -> q which means:
If the panda yawns, than the panda is alert.
The panda is not alert translates to:
q
Therefore, she yawned translates to:
p
You can put the q and the p together to form:
q -> p which means
If the panda is not alert, then she yawns.

What you have now is the statement of:
p -> q
and the converse of:
q -> p

Since the converse is not equivalent to the statement, then these 2 statements cannot be considered to be equivalent and you cannot assume that just because p -> q is true that q -> p is also true.

that's the fallacy of the converse.
It's the assumption that just because the statement is true that the converse of the statement is also true.

Here's an example where the converse of the statement is not true when the statement is true.

p = an animal is a pig
q = an animal is a mammal
p -> q means that "if an animal is a pig, then it is a mammal"
the converse to that statement is:
q -> p which means that "if an animal is a mammal, then it is a pig"
You can easily see that, just because a pig is a mammal, it does not necessarily mean that a mammal is a pig. A particular mammal may be a pig, but not all mammals are pigs.

You can put this solution on YOUR website!
i believe the answer depends on the values of x and y.

if you set z = sqrt(x^2 - y^2), you can square both sides of that equation to get:
z^2 = x^2 - y^2
if you add y^2 to both sides of that equation, then you get:
z^2 + y^2 = x^2 which can be shown as:
x^2 = z^2 + y^2

this looks a lot like the pythagorean formula of c^2 = a^2 + b^2, except:

x = c
z = a
y = b

c is the hypotenuse of a right triangle.
a and b are legs of the right triangle.

this can be translated to:

x is the hypotenuse of a right triangle.
z and y are legs of the right triangle.

is that what you're looking for?

if not, then what i the problem you are trying to solve?

You can put this solution on YOUR website!
34 is equal to 10 * 3 + 1 * 4
38 is equal to 10 * 3 + 2 * 4
48 is equal to 10 * 4 + 2 * 4
49 is equal to 10 * 4 + 2 * 4 + 1 * 1

is that what you're asking?

You can put this solution on YOUR website!
the statement is:
if i am hungry then i will eat
the inverse of that statement is:
if i am not hungry then i will not eat.
the converse of that statement is:
if i will eat, then i am hungry.
the contra-positive of that statement is:
if i do not eat, then i am not hungry.

we can translate those statements into logic symbols and variables.

we let p = i am hungry
we let q = i will eat

if i am hungry then i will eat translates to p->q
if i am not hungry then i will not eat translates to ~p->~q
if i eat, then i am hungry translates to q->p
if i don't eat, then i am not hungry translates to ~q->~p

the general rule is:
p->q is the statement.
~p->~q is the inverse.
q->p is the converse.
~q->~p is the contra-positive.

the statement and the contra-positive are equivalent. if the statement is true, then the contra-positive is also true.
the inverse and the converse are equivalent. if the inverse is true, then the converse is also true.

the statement and the converse are not equivalent. sometimes the converse is true when the statement is true and sometimes it's not.
the statement and the inverse are not equivalent. sometimes the inverse is true when the statement is true and sometimes it's not.

the original statement is:
if i am hungry then i will eat
the statement you are asked to analyze is:
if i am not hungry then i will not eat.
that's the inverse of the statement and it is not true by the fallacy of the inverse.

You can put this solution on YOUR website!
your truth table looks like this:

     p  q  r  ~p  ~r  (r->~p)  (q->~r)  (r->~p)^(q->~r)
     -  -  -  --  --  -------  -------  ---------------
     T  T  T  F   F      F        F            F
     T  T  F  F   T      T        T            T
     T  F  T  F   F      F        T            F
     T  F  F  F   T      T        T            T
     F  T  T  T   F      T        F            F
     F  T  F  T   T      T        T            T
     F  F  T  T   F      T        T            T
     F  F  F  T   T      T        T            T

if p is true then ~p is false.
if p is false, then ~p is true.
if r is true, then ~r is false.
if r is false, then ~r is true.

if r is false then r->~p is true, whether or not ~p is true.
the only time r->~p is false, is if r is true and ~p is false.

if q is false, then q->~r is true, whether or not ~r is true.
the only time q->~r is false is if q is true and ~r is false.

this is your truth table for the implied statement.

       A   B   A->B
       T   T     T
       T   F     F
       F   T     T
       F   F     T

you can see that A->B is only false when A is true and B is false.

when you apply the truth table for the implied statement, you assign different variables to A and B.
The A and B are just place holders to show you the logic.
for example:
A->B becomes r->~p after you replace A with r and B with ~p.

this is your truth table for the and statement.

       A   B    A^B
       T   T     T
       T   F     F
       F   T     F
       F   F     F

You can see that A^B is only true when A is true and B is true.
in your problem above:
A^B becomes (r->~p)^(q->~r) after you replace A with (r->~p) and you replace B with (q->~r).

in your problem there is a cascading of statements.
here's your truth table again for ease of reference.
only the fourth row is shown.

     p  q  r  ~p  ~r  (r->~p)  (q->~r)  (r->~p)^(q->~r)
     -  -  -  --  --  -------  -------  ---------------
     T  F  F  F   T      T        T            T        (fourth row)

it states that (r->~p)^(q->~r) is true.
this can only happen if (r->~p) is true and (q->~r) is true.
   (r->~p) is true if:
        r is false.
        r is true and ~p is true.
    this particular row has r is false.
    that's enough to make (r->~p) true.
(q->~r) is true if:
        q is false.
        q is true and ~r is true.
    this particular row has q is false.
    that's enough to make (q->~r) true.

once you have (r->~p) column and (q->~r) column, then you use those columns to find the truth of the and statement of (r->~p)^(q->~r).

You can put this solution on YOUR website!
you have three variables.
they are p, q, and r.
your compound statement is p ^ (q v ~r)
in english that would be:
p AND (q OR (not r))
here's your truth table:

           p   q   r   ~r   (q v ~r)   p ^ (q v ~r)
           -   -   -   --   --------   ------------
           T   T   T    F       T          T *****
           T   T   F    T       T          T *****
           T   F   T    F       F          F
           T   F   F    T       T          T *****
           F   T   T    F       T          F
           F   T   F    T       T          F
           F   F   T    F       F          F
           F   F   F    T       T          F

from this table, you can see that:
if p is true and (q or ~r) is true, then p ^ (q v ~r) is true.
in all other cases, p ^ (q v ~r) is false.

note that only one of q and ~r has to be true in order for the compound statement of (q v ~r) to be true.
if p is true, then the OR statement is true.
if ~q is true, then the OR statement is true.
The OR statement is only false if both p and q are false.

You can put this solution on YOUR website!
1. If possible evaluate g(t)for the given values of t
g(t)=2t^3 - t^2 + 4

you did not give any values of t so i wouldn't know how to answer question 1.
the procedure, however, would be the same as shown below:

equation g(t) = 2t^3 - t^2 + 4
solve for t = 3
the equation becomes:
g(3) = 2*(3)^3 - (3)^2 + 4
all you're doing is replacing t with 3 in the equation.
the answer, in that case, would be:
g(3) = 2*27 - 9 + 4 which becomes:
g(3) = 54 - 5 which becomes:
g(3) = 49


2. log(2,2^6)

you are looking to find the log of 2^6 to the base of 2.
since, in general, log(a^b) = b*log(a), your expression becomes:
6 * log(2,2)

to find 6 * log(2,2), you can use your calculator by converting the base of 2 to the base of 10 using the logarithm base conversion formula shown below.

6 * log(2,2) = 6 * ((log(10,2) / log(10,2))

the general formula is:

log(a,b) = log(c,b) / log(c,a)
a is the base you want to convert from.
c is the base you want to convert to.
b is the log value you are looking to solve for.
the expression of log of b to the base a is converted to:
log of b to the base c divided by log of a to the base c.

this allows you to use the LOG function of your calculator.

the formula becomes:

6 * log(2,2) = 6 * (LOG(2) / LOG(2)).

the answer is 6 * 1 because anything divided by itself is equal to 1.

your final answer is 6.

you would want to confirm your answer to make sure it's correct.
the basic definition of logs states:
y = log(b,x) if and only if b^y = x
your original expression is:
log(2,2^6)
set y equal to this to get:
y = log(2,2^6)
set x equal to 2^6 to get:
y = log(2,x)
set b equal to 2 to get:
y = log(b,x)
the basic definition of logarithms states that:
y = log(b,x) if and only if b^y = x
now, you know that b is equal to 2 and you know that x = 2^6, so this equation becomes:
2^y = 2^6
this equation is true if y = 6 because then you get:
2^6 = 2^6 which is true.
6 is the answer to your question.
log(2,2^6) = 6

You can put this solution on YOUR website!
449/.42 is equal to 1069.047619
the nearest tenth is the first digit after the decimal point.
since the second digit after the decimal point is equal to less than 5, you just truncate to the first digit after the decimal place.
your answer would be 1069.0

now if you do not want to perform the arithmetic (i used a calculator), then you might want to convert the fraction as follows:

multiply the fraction by 100/100 to get 44900/42
that's an equivalent fraction without a decimal point.
use your calculator to divide 44900 by 42 and you get 1069.047619.
this is the same answer as above, as it should be.

now if you don't want to do any of the above, but just want to translate the problem by dividing by a number to the nearest tenth rather than to the nearest hundredth as it's shown, then you simply truncate the .42 and make it .4 because the .02 is less than .05
if your number was .45 or above, then you would make it .5
since it's below .45, then you would make it .4

your expression would be 449/.4

if this doesn't answer your question, come back with a more detailed explanation of what you are looking for.