Question

Consider the following expressions:

(i) false

(ii) Q

(iii) true

(iv) P ∨ Q

(v) ¬ Q ∨ P

The number of expressions given above that are logically implied by P ∧ (P ⇒ Q) is ________

Answer 1

Concept:

P ⇒ Q means P̅ ∨ Q

Explanation:

P ∧ (P ⇒ Q) = P ∧ (P̅ ∨ Q) = P ∧ Q

Let us consider P.Q as A.

We have to find which of the given expression is implied by A, that is, (A => )

Construct the truth table for each option separately

Let A ≡ P ∧ (P ⇒ Q)

P	Q	P ⇒ Q	A
True	True	True	True
True	False	False	False
False	True	True	False
False	False	True	False

 

Consider all options one by one:

(i) False

P	Q	A	A ⇒ False
True	True	True	False
True	False	False	True
False	True	False	True
False	False	False	True

 

Here, all are not true. So, this option is incorrect.

(ii) Q

P	Q	A	A ⇒ Q
True	True	True	True
True	False	False	True
False	True	False	True
False	False	False	True

 

This option is implied by P ∧ (P ⇒ Q).

iii) True

P	Q	A	A ⇒ True
True	True	True	True
True	False	False	True
False	True	False	True
False	False	False	True

 

iv) P ∨ Q

P	Q	A	A ⇒ P ∨ Q
True	True	True	True
True	False	False	True
False	True	False	True
False	False	False	True

 

This is a tautology. So, option is correct.

v) –Q ∨ P

P	Q	A ≡ P ∧ (P ⇒ Q)	A ⇒ –Q ∨ P
True	True	True	True
True	False	False	True
False	True	False	True
False	False	False	True

 

This is also a tautology.

So, total 4 are implied by the given expression.