i am in my second year of math for the liberal arts and a problem has be terribly confused..

can you answer this

[(p --> q) ^ (q --> p)] --> (p <--> q)

is this an implication??

i would love to hear your answer

thank you...

danielle

Hi Danielle,

YES. The word IMPLICATION simply refers to the form of a statement: anything in the form "statement, arrow, statement" is an implication. For the purposes of logic, neither the left-hand statement nor the right-hand statement has to have any meaning, nor do they have to fit together in any way that makes any sense. They just have some "truth value" assigned. Your example happens to have meaning, but that is irrelevant to your question. As it turns out, in your example the left-hand statement has exactly the same truth value as the right-hand statement (no matter what meaning you attach to p and to q), so your implication happens to be a theorem. (It can be used as a definition of the "double arrow" notation.) To get an example that actually has meaning just let p be "n is an even number" and q be "n is divisible by 2".

Cheers,
Chris