GATE DISCRETE MATHEMATICS

Question

Show without using a truth table that 

                         p V q

                           ~ p

 ^{----------------------------------------------------------------}

                          ∴  q

Answer 1

The premises must be true. So, pVq is true and p' is true.

So, p is false. So, q must be true.

Hence, proved.

Answer 2

To show this we have to prove that ......

((p v q) ∧ ∽p )→q  is a tautology

Now this equals

∽((p v q) ∧ ∽p )∨ q

≅ (∽p ∧ ∽q) v p v q

Now distributing p over ∽p ∧ ∽q we get...

((p v ∽p)∧(p v ∽q )) v q

≅ p v ∽q v q

≅ p v 1

≅ 1

Hence it is a tautology