Explain, without using a truth table, why (p ? q ? r ) ? (¬ p ? ¬ q) is true when at least one of p, q , and r is true and at least one is false, but is false when all three variables have the same truth value.

Solution: Step-1: In this problem we need to explain without using a truth table , is true. Given: is true. When at least one of p , q, and r is true and at least one is false , but is false when all three variables have the same truth value. Conjunction : If p and q are statements , then the statement is true only when both p and q are true , and is false otherwise. Disjunction: If p and q are statements , then the statement is true when at least one of the two statements is true , and is false when both are false.Step-2: By using the above definitions : If at least one of p , q, and r is true if and only if the result is true. …………….(1) If at least one of p , q, and r is false if and only if the result is true. ……………..(2) From (1) , (2) it is clear that the entire statement is true if and only if there is at least one true and one false. So ,the given statement is false when all three variables have the same truth value.