Math Problem Statement
Use properties to show that (¬q ∧ (𝑝⇒q)) ⇒ ¬q is a tautology. You must clearly label each step of the logical equivalence. [1 mark]
Solution
To show that is a tautology, we need to prove that it is true in all possible cases using logical equivalences and properties. Let's break it down step by step, clearly labeling each step.
1. Express the implication using logical equivalence
Recall that an implication is equivalent to . So, we can rewrite the expression as follows:
2. Distribute the conjunction
Now, distribute the conjunction using the distributive property:
3. Simplify the conjunctions
Next, simplify the conjunctions:
- is a contradiction, so it simplifies to false.
- remains as it is since there's no contradiction.
4. Remove the false term
Since , we can remove the false term:
5. Express the implication
Now, we consider the whole expression . Using the previous steps, we now have:
An implication is false only when is true and is false. Let's analyze this:
- If is true, it means both and are true. But if is true, then the right-hand side is also true, making the implication true.
- If is false, the implication is vacuously true.
Conclusion:
In all cases, the implication is true, so it is a tautology.
Do you want any further details or explanations? Here are some related questions you could explore:
- What is a truth table, and how would it confirm this tautology?
- How do logical equivalences simplify logical expressions?
- What is the importance of the distributive property in logic?
- Can all implications be converted using ?
- What are some examples of contradictions in logic?
Tip: Understanding common logical equivalences like De Morgan's laws and implication rules helps in solving complex logical expressions easily!
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Math Problem Analysis
Mathematical Concepts
Propositional Logic
Tautology
Logical Equivalence
Formulas
Implication: p ⇒ q ≡ ¬p ∨ q
Distributive Property: ¬q ∧ (¬p ∨ q) ≡ (¬q ∧ ¬p) ∨ (¬q ∧ q)
Theorems
Logical Implication
Distributive Property
Contradiction
Suitable Grade Level
Undergraduate (Logic/Discrete Mathematics)