Question

Is the statement ((p V q) => r) <=> ((p=>r) ^ (q =>r)) a tautology, a contradiction, or neither?

Answer 1

So,
((p V q) => r) => ((p=>r) ^ (q =>r))
and,
((p V q) => r) <= ((p=>r) ^ (q =>r))

Answer 2

i think its a tautology

Answer 3

you think its always true?

Answer 4

I am not sure but if p is true and q is false, then left hand side is r and right hand side is not r.

Answer 5

i want to say neither because they are not equal

Answer 6

or maybe contradiction because how could it be.. if p or q then r iff if p then r and if q then r

Answer 7

If p and q are both true, then the statement is true.
If p is true and q is false, then the statement is false (not sure).

Answer 8

so that means it cant be tautology

Answer 9

this is a tautology

Answer 10

snap!

Answer 11

why @ganeshie8

Answer 12

Why? The left hand side is implying that p implies r or q implies r but the right hand side is saying that p implies r and q implies r at the same time?

Answer 13

just simplify left hand side and show that it equals rigth hand side

Answer 14

(p V q) => r

is logically same as

p'q' + r

which is same as

(p'+r)(q'+r)

which is same as right hand side

Answer 15

but you still have a difference between the or and the and sign dont you?

Answer 16

I am using below :

```
a => b

is logically same as

a' + b
```

Answer 17

so it is always true, wow

Answer 18

Yep! create a truth table maybe

http://www.wolframalpha.com/input/?i=truth+table++%28%28%28p+or+q%29+implies+r%29+%5C%5BEquivalent%5D++%28%28p+implies+r%29+and+%28q+implies+r%29%29%29

Answer 19

@thomas5267 it seems "implication" is not distributive over "or"