SOLUTION: 8. Determine which, if any, of the three statements are equivalent. Show steps. Give a reason for your conclusion. Show complete work and submit your solution to the Dropbox. 1)

Algebra ->  Conjunction -> SOLUTION: 8. Determine which, if any, of the three statements are equivalent. Show steps. Give a reason for your conclusion. Show complete work and submit your solution to the Dropbox. 1)      Log On


   

Question 548454: 8. Determine which, if any, of the three statements are equivalent. Show steps. Give a reason for your conclusion. Show complete work and submit your solution to the Dropbox.
1) if it is sunny, then I will go swimming
2) If i do not go swimming, then it is not sunny.
3) Either it is sunny or I will go swimming.
a)1,2,and 3 are equivalent
b)1 and 2 are equivalent
c)2 and 3 are equivalent
d)1 and 3 are equivalent
e) none are equivalent
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
The drop box! I thought I had seen the last of it when my son graduated from Connections Academy.
It is obvious to me that 1 and 2 are equivalent. Statement 3 is not equivalent to them. So the right answer is b).
How to prove it to your teacher's content is another story.
Your lessons or textbook must give you the right format for your proof.
A=it's sunny
B=I go swimming.
Statement 1 says If A, then B.
Statement 2 says If not B, then not A. That is the contrapositive of statement 1, and they are both equivalent.
Statement 3 says Either A or B is true, which I take to imply that both cannot be true at the same time.
They may have taught you ways to express the statements and symbols to abbreviate
the "If ...,then ..." relationship and the "not" in the statements.
They would have told you that a statement implies its contrapositive. If one is true, so is the other.
Alternatively you could have been told to use diagrams to express relationships.
If A, then B can be represented as
The inner circle would represents "A is true" (It's sunny).
Every point in that circle is certainly in circle B.
The large circle,B , represent "B is true" (I go swimming).
I may go swimming when it's not sunny, so those cases would be in the part of circle B that is not inside circle A.
You can graphically see that if not B (if you are outside circle B), not A (you are outside of circle A too.
Either A or B (but not both) , meaning "Either it is sunny or I will go swimming." would be represented by
You cannot be in both circles at the same time.