prove the following equivalence without constructing truth tables. Use only equi

Question

prove the following equivalence without constructing truth tables. Use only equivalence statements. in your proofs a) (p v q) ---> r equivalent to ~r --->(~p ^ ~q) b) (p^q)----->r equivalent to ~r---> (~p v ~q)

Explanation / Answer

we have the equivalence statement that p ---> q is equivalent to ~q ----> ~p and ~(p v q) = ~p ^ ~q , ~(p ^ q) = ~p v ~q a) (p v q) ---> r is equivalent to ~r ----> ~(p v q) ~r ----> ~(p v q) ==> ~r ----> (~p ^ ~q) so (p v q) ---> r equivalent to ~r --->(~p ^ ~q) hence proved b) (p^q)----->r is equivalent to ~r ---> ~(p ^ q) ~r ---> ~(p ^ q) ==> ~r ---> (~p v ~q) so (p^q)----->r equivalent to ~r---> (~p v ~q) hence proved

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