Question

How to show something is not logically equivalent without using truth tables?

Answer 1

Like (p ^q ) -->r is not equivalent with (p -->r) ^(q -->r)

Answer 2

there are rules that have been determined by the use of truth tables to cut down on the amount of re wheel inventing that goes on

Answer 3

(p^q) -> r

-(p^q) v r

-p v -q v r

--------------------

(p -> r) ^ (q ->r)

(-p v r) ^ (-q v r)

[ -p^ (-q v r) ] v [ r ^ (-q v r) ]

(-p^-q) v (-p^r) v (r ^-q) v (r^ r)

(-p^-q) v (-p^r) v (r ^-q) v r

but sometimes you just tend to go around in circles

Answer 4

if u think the statement might be true, reduce it till u got somethings that its a theorem already proved or an axion. To prove its false just find a counter example.

For example:

(p^q) -> r == (p -> r) ^ (q ->r)

if p is true, q false and r false then:

true ^ false -> false == true -> false ^ false -> false
false -> false == false ^ true
true == false
false

there u got a counter example and just that its enough to prove its not equal

Answer 5

Thank you, fedep3 I think that's an excellent method rather than simplifying but thanks for everyones help!