## Discrete Mathematics with Applications 4th Edition

Let p represent "2 is a factor of n." Let q represent "3 is a factor of n." Let r represent "6 is a factor of n." "If 2 is a factor of n and 3 is a factor of n, then 6 is a factor of n" in symbolic form is p $\land$ q $\rightarrow$ r. "2 is not a factor of n or 3 is not a factor of n or 6 is a factor of n" in symbolic form is ~p $\lor$ ~q $\lor$ r. These two statements are logically equivalent. See the truth table:
To construct the truth table, first fill in the 8 possible combinations of truth values for p, q, and r. To evaluate p $\land$ q recall the definition of AND (a $\land$ b is true when, and only when, both a and b are true. If either a or b is false or if both are false, a $\land$ b is false). To evaluate p $\land$ q $\rightarrow$ r, recall by the definition of a conditional statement, when the if element is T and the then element is F, the statement is F. In all other cases the statement is T. To evaluate ~p $\lor$ ~q $\lor$ r recall the definition of OR (a $\lor$ b is true when either a is true, or b is true, or both a and b are true; it is false only when both a and b are false). The two statements are only logically equivalent if, and only if, they have identical truth values for each possible substitution of statements for their statement variables.