Equivalent Negations
A free Logic lesson from the “Logical Equivalence” unit, with a worked example and practice problems including step-by-step solutions.
Equivalent statements agree on every assignment. Use truth tables or justified identities, not similar-looking wording. De Morgan swaps and/or and negates both parts. The implication p→q equals ¬p∨q. One disagreeing row disproves a claimed equivalence.
What you'll learn
- Rewrite negations without changing meaning
- Explain the idea in plain English before using symbols
- Use examples, non-examples, or counterexamples to check the reasoning
Worked example
Problem. Rewrite ¬(p ∧ q) using De Morgan's Law.
- Negate both parts and switch and to or.
Answer: ¬p ∨ ¬q
Lesson preview: this page shows 5 practice items. The interactive activity contains 6 practice items and 6 quiz items (multiple choice). Counts include repeated prompts and variants.
Practice problems
1. Rewrite ¬(p ∧ q) using De Morgan's Law.
Choices: p ∨ q · p ∧ q · ¬p ∨ ¬q · ¬p ∧ ¬q
Show solution
- Negate both parts and switch and to or.
Answer: ¬p ∨ ¬q
2. Rewrite p→q without an implication symbol.
Choices: ¬p ∧ q · ¬p ∨ q · p ∨ ¬q · p ∧ q
Show solution
- Both expressions are false exactly in the p=true, q=false row.
Answer: ¬p ∨ q
3. Simplify p ∧ (p ∨ q).
Choices: p · q · p ∧ q · ¬p
Show solution
- If p is false, the expression is false.
- If p is true, both factors are true regardless of q.
Answer: p
4. Which assignment shows p→q is not equivalent to q→p?
Choices: p=true, q=true · p=false, q=false · No such assignment exists · p=false, q=true
Show solution
- At this assignment p→q is true, while q→p is false.
Answer: p=false, q=true
5. Simplify ¬(¬p ∨ ¬q).
Choices: ¬p ∧ ¬q · p → q · p ∧ q · p ∨ q
Show solution
- De Morgan gives ¬¬p ∧ ¬¬q.
- Cancel the double negations.
Answer: p ∧ q
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