Prove that R is an equivalence relation. Show that it satisfies all the properties you selected in part (a), and submit your proof as a free response.

Question

Let A be the set of all statement forms in the three variables p, q, and r, and let R be the relation defined on A as follows.

For all S and T in A, S R T  ⇔  S and T have the same truth table.

Prove that R is an equivalence relation. Show that it satisfies all the properties you selected in part (a), and submit your proof as a free response.

Answer 1

REFLEXIVE property....

S ~ S means that S has the same truth table as S...

T ~T means that T has the same truth table as T...

Of course!!!

SYMMETRIC property...

S ~T means that S has the same truth table as T.

Then T has the same truth table as S.

So T ~ S

TRANSITIVE Property...

S~T and T~V and V ~W

S has the same truth table T AND T has truth table V And V has the same truth table as W

So then S,T,V,W all have the same truth table...

In particular, S has the truth table as W

Then S~T~V~W ---> S~W