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Discrete Math Help - Truth Tables?
My task is to prove that the Original Statement (P => Q) is logically equivalent to the Contrapositive Statement (!P => !Q) using Truth Tables.
It seems like you would come out with...
P=>Q !P=>!Q
T T
F T
T F
T T
... which aren't equivalent. Or maybe they are, I'm not sure.
How are you supposed to go about proving this?
1 Answer
- ?Lv 67 years agoFavorite Answer
I don't think they are but lets look at the meaning of logically equivalent.
A single arrow between P and Q means that if P then Q but not, if Q then P. Hence P-->Q is not logically equivalent to Q-->P.
An example of this would be where P is being located in a city in the US say New York, and Q is being located in the US. Hence If located in New York, then you are located in the US (P-->Q)
But if in the US, one is not necessarily in New York. So Q-->P is not necessarily true.
Applying this to, if not in New York then not in the US, is also not necessarily true (you could be in another US city). Hence P-->Q is not logically equivalent to !P-->!Q.
TT
FT
FF
As you can see from the possible truth combinations, Not P can result in both outcomes for Q.
