The proposition ( ~p) ∨(p∧ ~q) is equivalent to:

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p→

p∨

q→p

p∧

answer is A.

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Detailed Solution

Concept: We know from the mathematical logic that if the statement pp has a truth value T or F then the negation of pp is denoted as ~p and has truth value F or T respectively.
We also know that when there are two statements p and q, the statement with a conjunction (with logical connective AND) of their truth values is denoted as p∧q and has a truth value T only when both pp and q have truth values, T, otherwise false. The statement with disjunction (with logical connective OR) of their truth values are denoted as p ^q and has a truth value T only when one of p and q have truth value T, otherwise false.
The statement with the implication (with logical connective If...then...) of their truth values is denoted as p→q and has a truth value F only when one of p has a truth value T and q has a truth value F otherwise true.
We know that two composite statements are equivalent when they have the same truth value for all possible combinations of truth values for all prime statements appearing in the two composite statements.
We are asked to find the equivalent statement of ( ~p) ∨(p∧ ~q) So let us draw its truth table for ( ~p) ∨(p∧ ~q) ,p→ ~q,p∨ ~q,q→p,p∧ ~q

p	q	~p	~q	p→ ~q	p∨ ~q	q→p	p∧ ~q	( ~p) ∨(p∧ ~q)
T	T	F	F	F	T	T	F	F
T	F	F	T	T	T	T	T	T
F	T	T	F	T	F	F	F	T
F	F	T	T	T	T	T	F	T

We see that the truth values of in the column ( ~p) ∨(p∧ ~q) matches only with truth values in the column of p→ ~q. So they are equivalent. The composite statement p→ ~q
Hence, option 1 is correct.

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