Using Logical simplification

Question

Could someone help me on the question below please.

 

Using Logical simplification, prove that the following are tautologies, contradictions or neither. Verify your answers using a truth tables.

(p ⇒ q) ∨ (q ⇒ p)
(p∨((p’∨ q)∧ q’))
(p∧((p’∨ q)∧ q’))
((p ⇒ q) ⇒ p) ⇒ p

Answer 1

This online system doesn't handle tables well, but I'll get you started on the first one and you can verify it with a truth table if you wish. For all of these you are just going to use various rules of implication and replacement to try to reduce them to a simpler, clearer form.

 

(p -> q) v (q -> p)

(~p v q) v (~q v p) Material implication

(p v ~p) v (q v ~q) Commutativity and Associativity

T v T

T

 

And right there you can see that the truth value of this expression is always true or 1. Depending on the precise definition you are working with, this would generally qualify as a tautology, or a logical expression that always evaluates as true.

 

You'll need to do something like this for each expression -- apply the various rules of inference -- to transform the argument so that its truth value is clear.