Question 191839:
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website! # 1
To write any set in roster form, simply write out EVERY element that is in the set. Since "Set N is the set of natural numbers between four and eleven.", this means we need to write out the set of natural (ie whole) numbers from 4 to 11 like so:
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# 2
Take note how every element in the set
is a whole number. So this suggests that they are natural (ie counting) numbers. In addition, the interval we'll be working with is  where "x" is a number in the set
Putting this altogether, we get the compacted set written in set-builder notation:
Note: this reads: "the set Z is comprised of x such that x is a natural number and x is in between 9 and 17".
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# 3
First, let's write out EVERY element that is in set A (ie let's write set A in roster form)
Now looking at the elements of the set  take note how EVERY element of this set is in set A. So this shows us that set B is a subset of set A.
However, since the element "19" is in A but NOT in B, this means that set A is NOT a subset of set B. This means that set A does NOT equal set B
So we've determined that set B is a subset of set A.
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# 4
a)
Recall, that  is the intersection of sets P and Q. This intersection is a set of ALL of the common elements. Visually, the intersection lies in the overlapping region of a venn diagram (for 2 circles).
Looking at the drawing
We can see that the elements "B" and "E" lie in the overlapping region. So this simply means that 
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b)
Also, remember that  is a set of the combination of sets P and Q (with the duplicates removed)
From the drawing, we can determine that
 and 
Note: it might help to partially erase one set to get a good look at the other.
Now combine the two sets to get
Note: ALL of the elements of lie in either circle or both circles. So all you really need to do is list the elements that lie in the circles.
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# 5
U = {21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31},
A = {21, 23, 25, 27, 29}, and
B = {22, 25, 28, 29}.
First, let's find the set A'. So form the set of elements that are in U but NOT in A:
Do the same for B'. List the elements in U but NOT in B:
Now to find  , just combine the two sets A' and B' (and remove duplicate elements) to get
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# 6
Given U = {l, m, n, o, p, q, r, s, t, u, v, w}, A = {l, o, p, q, s, t}, B = {n, o, r, s, v, w},
and C = {l, m, n, q, r, t}, find (A' U C') n B '.
A' = {m, n, r, u, v, w}
B' = {l, m, p, q, t, u}
C' = {o, p, s, u, v, w}
First, let's find A'. So form a set of elements from U but NOT in A:
Now let's find B'. Apply the same technique but form a set of elements from U but NOT from B:
Finally, do the same for C'. Make a set from U but NOT from C:
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Now let's find  :
Combine the sets A' and C' and remove duplicates to get
Now take the common elements from set and B' to get the elements: m, p, and u
Note: these elements are in BOTH sets and B'
So  \cap B' = \left\{m, p, u\right\})
So the final set we're looking for is
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# 7
Since "Set K contains 50 elements", we can write this in a more compact notation as  = 50) (think of this as "n"umber of elements in set K)
Also, because "set J contains 66 elements", we can write  = 66)
and finally, since "14 elements are common to both sets", we can say that =14)
Now it turns out that the number of elements in the union is related to the number of elements of both sets and the number of elements in their intersection. This value can be found by the formula:
 = n(K) + n(J) - n\left(K \cap J\right))
All we're doing here is "combining" the sets by adding the number in each set and subtracting out any repeated values (which happen to lie in the intersection)
... Start with the given formula
 = 50 + 66 - 14) ... Plug in , , and
 = 116 - 14) ... Add 50 and 66 to get 116.
 = 102) ... Subtract 14 from 116 to get 102.
So the answer is which means that there are 102 elements in both sets (with duplicates removed)
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# 8
Given A = {1, 2, 3, 4}, B = {3, 4, 5, 6,}, and C = {4, 6, 7}.
a)
Take note that the common elements between sets A and B are the elements "3" and "4", so this means that:
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b)
The only common element between set A and set C is the element "4". So...
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c)
Combine the sets A and C (and remove duplicates) to form:
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d)
Similarly, combine sets B and C and get rid of any repeated elements to get:
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e)
First, we need to find  . So combine sets A and B to get
Now let's look for the common elements between and C. Since the elements "4" and "6" are in both sets, this means
 \cap C = \left\{4, 6\right\})
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f)
Using the set from part d), simply combine this set with set A to get:
 = \left\{1, 2, 3, 4, 5, 6, 7\right\})
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g)
From part a), we found that . Since the only common element between this set and set C is the element "4", this tells us that
 \cap C = \left\{4\right\})
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h)
Again from part a), we know that . Combine this set with set C (and remove repeated values) to get the new set:
 \cup C = \left\{3, 4, 6, 7\right\})
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# 9
To start things off, let's draw three blank overlapping circles and label them "car", "refrigerator", and "washer". Now draw a large rectangle that surrounds the three circles and label this rectangle U (for the universal set).
Since "Three consumers indicated that they were going to buy all three items", this means that the value "3" goes in the direct center of the overlapping circles.
Now because "9 were going to buy a washer and a refrigerator", this means that some of those 9 people might have bought all three. So to find the ones who ONLY bought a washer and a refrigerator (not all three), we need to subtract 3 from 9 to get:  . So this tells us that 6 people bought a washer and a refrigerator (and not all three). So place this value in the overlapping region between the circles labeled "refrigerator" and "washer".
Also, since "15 were going to buy a car and a washer" and we just want to find those who ONLY bought a a car and a washer (not all three), we just subtract 3 from 15 to get  . So 12 people bought a car and a washer (not all three). Now place this value in the region between the "car" and "washer" circles.
Since "7 were going t o buy both a car and a refrigerator", and we're only interested in those who ONLY bought the two (not all three), we subtract 3 from 7 to get  . So there are 4 people who bought a car and a refrigerator (not all three). Stick this value in the region between the "car" and "refrigerator" circles.
So we should have the following so far:
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Now because "34 said they were going to buy a new washer" and we're only interested in the number who ONLY bought a washer (not a car or refrigerator), we need to subtract the number who bought all three (3 people), the number who bought a car and washer (12 people), and the number of people who bought a refrigerator and washer (6 people) from 34 to get  . So there are 13 people who ONLY bought a washer. Place this value in the "washer" circle (and not in any overlapping regions)
If the method above is a bit cumbersome, then let's try another method in finding the number of people who ONLY bought a car:
Since 12 people bought a car and a washer, 4 people bought a car and a refrigerator, and 3 people bought all three, this means that  people bought AT LEAST 2 items (where one of the items is a car). Since we ONLY want the car, we need to subtract this from 33 (the number people who bought a car and maybe something else) to get:  . So 15 people ONLY bought a car. Place this value in the "car" circle (and not in any overlapping regions)
Because 4 people bought a refrigerator and a car, 6 people bought a refrigerator and a washer, and 3 bought all three, this means that  people bought AT LEAST two items in which one item is a refrigerator. Subtract this value from 18 (the number who bought a refrigerator and maybe something more) to get  . So 5 people bought ONLY a refrigerator. Place this value in the "refrigerator" circle (and not in any overlapping regions)
Finally, add up EVERY element that is in every region to get:  . So 58 people bought AT LEAST one item (either a car, refrigerator, or washer). Subtract this value from the number who participated (100 people) to get  . So 42 people did NOT buy any of the items listed. Place this value outside the three circles but inside the set U (the rectangle).
So we should now have the completed Venn diagram:
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Now let's answer the questions:
a)
How many were going to buy only a car?
Looking at the diagram (and going over our previous work), we get 15 people only bought a car.
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b)
How many were going to buy only a washer?
Similarly, we see from the Venn diagram that 13 people only bought a washer.
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c)
How many were going to buy only a refrigerator?
From the diagram, we see that 5 people only bought a refrigerator.
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d) How many were going to buy a car and a washer but not a refrigerator?
From the diagram, we see that the number that lies in the region between the "car" and "washer" is 12. So 12 people bought a car and a washer but not a refrigerator.
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e) How many were going to buy none of these items?
Since the value 42 is outside all of the circles, this means that 42 people did not buy any of these items.
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# 10
First, draw a Venn diagram and place the value 14 (the number who had both) in the overlapping region. Also, write the value 27 (the number who had neither) in the region outside the circles.
Since there were 35 people with brown eyes and 14 of them had both brown eyes AND blonde hair, this means that 35 - 14 = 21 of them ONLY had brown eyes. So place this value in the "brown eyes" circle.
Also, because there were 24 people with blonde hair and 14 had both, subtract 14 from 24 to get 24-14=10. So 10 people ONLY had blonde hair. Place this value in the "blonde hair" circle.
So we have the following Venn diagram
Now just add up the values 14, 27, 21, and 10 to find the total number interviewed:
In other words, we're finding:
Total number of people interviewed = number of people who ONLY had brown eyes + number of people who ONLY had blonde hair + number of people who had both brown eyes AND blonde hair + number of people who had neither feature
Total number of people interviewed = 14 + 27 + 21 + 10 = 72
So a total of 72 people were interviewed
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# 11
Starting from the fact that 9 people ate all types, we know that since 13 ate both beef AND fish, this means that 13 - 9 = 4 people ate both beef AND fish but NOT poultry.
Also, we know that 14 ate both fish AND poultry. So, this means that 14 - 9 = 5 people ate both fish AND poultry but NOT beef.
Furthermore, we're told that 15 people at both beef AND poultry. So 15 - 9 = 6 people ate both beef AND poultry but NOT fish.
Now to find out how many people ate ONLY beef, simply subtract the number of people ate beef and other products from the number of people who ate beef like this:
People who ONLY ate beef = People who ate beef - people who ate beef AND fish - people who ate beef AND poultry - people who ate all three
People who ONLY ate beef = 25 - 4 - 6 - 9 = 6
So 6 people ONLY ate beef.
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Do the same with people who ONLY ate fish:
People who ONLY ate fish = People who ate fish - people who ate beef AND fish - people who ate fish AND poultry - people who ate all three
People who ONLY ate fish = 28 - 4 - 5 - 9 = 10
So 10 people ONLY ate fish
Finally, apply the same technique to find the number of people who ONLY ate poultry
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People who ONLY ate poultry = People who ate poultry - people who ate beef AND poultry - people who ate fish AND poultry - people who ate all three
People who ONLY ate poultry = 30 - 6 - 5 - 9 = 10
So 10 people ONLY ate poultry
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Once you have all of this information, draw the venn diagram and fill in the appropriate regions to get:
The following answers will use the venn diagram:
a)
Number of people who ate fish, but not poultry or beef = number of people who ONLY ate fish
Number of people who ate fish, but not poultry or beef = 10
So there were 10 people who ate fish, but not poultry or beef
b)
Number of people who ate poultry and fish, but not beef = number of people who ate both poultry AND fish (but not all three)
Number of people who ate poultry and fish, but not beef = 5
c)
Number of people who ate ONLY beef = 6
d)
Number of people who ate nothing or only poultry = number of people who ate nothing + number of people who ate poultry
Number of people who ate nothing or only poultry = 7 + 10 = 17
So there were 17 people who ate nothing or only poultry.
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