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. Expert's answer Let us explain, without using a truth table, why (p ∨ q ∨ r) ∧ (¬p ∨ ¬q ∨ ¬r)(p∨q∨r)∧(¬p∨¬q∨¬r) is true when at least one of p,p, q,q, and rr 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,p,q,q, and rr is true then the disjunction (p ∨ q ∨ r)(p∨q∨r) is true. If at least one of p,p,q,q, and rr is false then the disjunction (¬p ∨ ¬q ∨ ¬r)(¬p∨¬q∨¬r) is true. Therefore, in this case the conjunction (p ∨ q ∨ r) ∧ (¬p ∨ ¬q ∨ ¬r)(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)(¬p∨¬q∨¬r) is false, and hence the conjunction (p ∨ q ∨ r) ∧ (¬p ∨ ¬q ∨ ¬r)(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)(p∨q∨r) is false, and hence the conjunction (p ∨ q ∨ r) ∧ (¬p ∨ ¬q ∨ ¬r)(p∨q∨r)∧(¬p∨¬q∨¬r) is false.
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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.
Let us explain, without using a truth table, why is true when at least one of and 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 and is true then the disjunction is true. If at least one of and is false then the disjunction is true. Therefore, in this case the conjunction is true.
If all three variables have the same truth value equal to true then the value of disjunction is false, and hence the conjunction is false. In the case when all three variables have the same truth value equal to false then the value of disjunction is false, and hence the conjunction is false.