Explain, without using a truth table , why (p ∨ q ∨ r) ∧ (¬p ∨ ¬q ∨ ¬r) 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.

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Let us explain, without using a truth table, why (p ∨ q ∨ r) ∧ (¬p ∨ ¬q ∨ ¬r) 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.

If at least one of p, q and r is true then the disjunction (p ∨ q ∨ r) is true. If at least one of p, q and r is false, then the disjunction (¬p ∨ ¬q ∨ ¬r) is true. Therefore, in this case the conjunction (p ∨ q ∨ r) ∧ (¬p ∨ ¬q ∨ ¬r) is true.

If all three variables have the same truth value equal to true then the value of disjunction (¬p ∨ ¬q ∨ ¬r) is false, and hence the conjunction (p ∨ q ∨ r) ∧ (¬p ∨ ¬q ∨ ¬r) is false.

In the case when all three variables have the same truth value equal to false then the value of disjunction (p ∨ q ∨ r) is false, and hence the conjunction (p ∨ q ∨ r) ∧ (¬p ∨ ¬q ∨ ¬r) is false.

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