For statements $p$ and $q$, show that $(p \wedge \neg q) \wedge(p \wedge q)$ is a contradiction.
Answer (1)
Use associative and commutative properties to rearrange the expression: ( pw e d g ee g q ) w e d g e ( pw e d g e q ) e q u i v pw e d g e pw e d g ee g qw e d g e q .
Simplify pw e d g e p to p , resulting in pw e d g e ( e g qw e d g e q ) .
Recognize that ¬ qw e d g e q is always false.
Conclude that pw e d g e F a l se is always false, thus the original expression is a contradiction: F a l se .
Explanation
Understanding the Problem We are given the logical expression ( pw e d g ee g q ) w e d g e ( pw e d g e q ) and we want to show that it is a contradiction. A contradiction is a statement that is always false, regardless of the truth values of its components.
Simplifying the Expression We can use the associative property of the ∧ (AND) operator to regroup the expression as follows: ( pw e d g ee g q ) w e d g e ( pw e d g e q ) e q u i v pw e d g ee g qw e d g e pw e d g e q Since the AND operator is commutative, we can rearrange the terms: pw e d g e pw e d g ee g qw e d g e q Since pw e d g e p e q u i v p , we can simplify the expression to: pw e d g e ( e g qw e d g e q )
Analyzing the Contradiction Now, let's analyze the term ¬ qw e d g e q . This term represents the conjunction of a statement q and its negation ¬ q . By definition, if q is true, then ¬ q is false, and if q is false, then ¬ q is true. Therefore, ¬ qw e d g e q is always false, because it is impossible for both q and its negation to be true simultaneously. So, ¬ qw e d g e q e q u i v F a l se .
Concluding the Contradiction Substituting F a l se for ¬ qw e d g e q in our expression, we get: pw e d g e F a l se The conjunction of any statement with F a l se is always F a l se . Therefore, pw e d g e F a l see q u i v F a l se .
Final Answer Thus, the original expression ( pw e d g ee g q ) w e d g e ( pw e d g e q ) simplifies to F a l se , which means it is a contradiction.
Examples
In digital circuit design, contradictions are useful for identifying errors or impossible states in a system. For example, if a circuit's logic leads to a situation where a signal must be both high and low at the same time, this indicates a design flaw that needs to be corrected. Recognizing and eliminating contradictions ensures the reliability and correctness of digital systems.
