Question

The proposition $\left( \tilde{\ }p \right)\vee \left( p\wedge \tilde{\ }q \right)$ is equivalent to\[\]
A. $p\to \tilde{\ }q$ \[\]
B. $p\vee \tilde{\ }q$\[\]
C. $q\to p$\[\]
D. $p\wedge \tilde{\ }q$\[\]

Answer 1

Hint: We recall the basic operation of truth values of negation (Logical NOT ) conjunction (logical AND), disjunction (Logical OR), and implication (logical if-else ). We draw the truth table for the given statement and for each composite statement in the options. We compare results in columns to see if they have the same truth values. \[\]

Complete step by step answer:
We know from the mathematical logic that if the statement $p$ has a truth value T or F then the negation of $p$ is denoted as $\tilde{\ }p$ 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 $p\wedge q$ and has a truth value T only when both $p$ and $q$ have truth values, T, otherwise false. The statement with disjunction (with logical connective OR) of their truth values are denoted as $p\hat{\ }q$ 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 $p\to q$ 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 $\left( \tilde{\ }p \right)\vee \left( p\wedge \tilde{\ }q \right)$. So let us draw its truth table for $\left( \tilde{\ }p \right)\vee \left( p\wedge \tilde{\ }q \right),p\to \tilde{\ }q,p\vee \tilde{\ }q,q\to p,p\wedge \tilde{\ }q$\[\]

$p$	$q$	$\tilde{\ }p$	$\tilde{\ }q$	$p\to \tilde{\ }q$	$p\vee \tilde{\ }q$	$q\to p$	$p\wedge \tilde{\ }q$	$\left( \tilde{\ }p \right)\vee \left( p\wedge \tilde{\ }q \right)$
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 $\left( \tilde{\ }p \right)\vee \left( p\wedge \tilde{\ }q \right)$ matches only with truth values in the column of $p\to \tilde{\ }q$. So they are equivalent. The composite statement $p\to \tilde{\ }q$ is in option A. So the correct option is A\[\]
Note:
We note that the other equivalent statements are $q\to p,p\vee \tilde{\ }q$.We note that if the composite statement is always true then it called a tautology and if the composite statement is always false it is called a fallacy. The statement $\tilde{\ }q\to \tilde{\ }p$ is the contra-positive of $p\to q$ and $q\to p$ is the converse of $p\to q$.