Decide whether or not the following pairs of statements are

Decide whether or not the following pairs of statements are logically equivalent.

a) (P Q) R and ((P Q) R)

b) (P Q) and P Q

c) P (Q Q) and ( P) (Q Q)

Solution

Prepare truth table for each.

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b) ~(p implies q) and p ^ ~q

c) P (Q Q) and ( P) (Q Q)

Here Q or ~Q is a tautology.

Hence left side is only P and ~P which is always false.

Right side is also always false as Q and ~Q

Hence logically equivalent.