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De Morgan's Laws

A free Logic lesson from the “Logical Equivalence” unit, with a worked example and practice problems including step-by-step solutions.

De Morgan's Laws explain how not moves across and/or statements: not (p and q) becomes not p or not q; not (p or q) becomes not p and not q. Learning objective: Use De Morgan's Laws to negate and/or statements. Prerequisite: No formal prerequisite. Work in this lesson starts with ordinary language, then connects the idea to symbols only after the meaning is clear. Example 1: ¬(p ∧ q) is equivalent to ¬p ∨ ¬q. Example 2: A conditional is equivalent to its contrapositive, not necessarily to its converse. A common misconception is to treat familiar wording as proof; instead, check exactly what the statement says and what follows from it.

What you'll learn

Why it matters: Equivalent statements let students rewrite claims without changing meaning, a key habit in algebra and proof.

Worked example

Problem. Example case A (De Morgan's Laws): Worked example: Using De Morgan's Law, rewrite ¬(p ∧ q).

  1. Worked Example: First identify exactly what the question is asking: Example case A (De Morgan's Laws): Worked example: Using De Morgan's Law, rewrite ¬(p ∧ q).
  2. Use De Morgan's Laws: negating an and statement changes it to or, and negating an or statement changes it to and.
  3. De Morgan's Law switches ∧ to ∨.
  4. Both parts are negated.

Answer: ¬p ∨ ¬q

Practice problems

1. Practice case A (De Morgan's Laws): Practice: Using De Morgan's Law, rewrite ¬(p ∧ q).

Choices: ¬p ∨ ¬q · ¬p ∧ ¬q · p ∨ q · p ∧ ¬q

Show solution
  1. Warm-up: First identify exactly what the question is asking: Practice case A (De Morgan's Laws): Practice: Using De Morgan's Law, rewrite ¬(p ∧ q).
  2. Use De Morgan's Laws: negating an and statement changes it to or, and negating an or statement changes it to and.
  3. De Morgan's Law switches ∧ to ∨.
  4. Both parts are negated.
  5. So ¬(p ∧ q) becomes ¬p ∨ ¬q.
  6. Verify the selected choice by checking that it preserves the stated logical meaning and that the other choices change the rule or claim.

Answer: ¬p ∨ ¬q

2. Practice case B (De Morgan's Laws): Practice: Using De Morgan's Law, rewrite ¬(p ∨ q).

Choices: ¬p ∧ ¬q · ¬p ∨ ¬q · p ∧ q · p → q

Show solution
  1. Warm-up: First identify exactly what the question is asking: Practice case B (De Morgan's Laws): Practice: Using De Morgan's Law, rewrite ¬(p ∨ q).
  2. Use De Morgan's Laws: negating an and statement changes it to or, and negating an or statement changes it to and.
  3. De Morgan's Law switches ∨ to ∧.
  4. Both parts are negated.
  5. So ¬(p ∨ q) becomes ¬p ∧ ¬q.
  6. Verify the selected choice by checking that it preserves the stated logical meaning and that the other choices change the rule or claim.

Answer: ¬p ∧ ¬q

3. Practice case C (De Morgan's Laws): Practice: Which statement is equivalent to ¬(p ∧ q)?

Choices: ¬p ∨ ¬q · ¬p ∧ ¬q · p ∨ q · p ∧ q

Show solution
  1. Equivalent statements match in every row.
  2. De Morgan gives ¬(p ∧ q) = ¬p ∨ ¬q.
  3. The connective flips and both parts negate.

Answer: ¬p ∨ ¬q

4. Practice case D (De Morgan's Laws): Practice: Which statement is equivalent to ¬(p ∨ q)?

Choices: ¬p ∧ ¬q · ¬p ∨ ¬q · p ∧ q · p ↔ q

Show solution
  1. Equivalent statements match in every row.
  2. De Morgan gives ¬(p ∨ q) = ¬p ∧ ¬q.
  3. The connective flips and both parts negate.

Answer: ¬p ∧ ¬q

5. Practice case E (De Morgan's Laws): Practice: Which is an INCORRECT rewrite of ¬(p ∧ q)?

Choices: ¬p ∧ ¬q · ¬p ∨ ¬q · not (p and q) · ¬q ∨ ¬p

Show solution
  1. Core Practice: First identify exactly what the question is asking: Practice case E (De Morgan's Laws): Practice: Which is an INCORRECT rewrite of ¬(p ∧ q)?
  2. Use De Morgan's Laws: negating an and statement changes it to or, and negating an or statement changes it to and.
  3. The correct rewrite is ¬p ∨ ¬q.
  4. Leaving the connective as ∧ is the classic mistake.
  5. So ¬p ∧ ¬q is incorrect.
  6. Verify the selected choice by checking that it preserves the stated logical meaning and that the other choices change the rule or claim.

Answer: ¬p ∧ ¬q

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