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Equivalent Conditional Statements

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

A conditional and its contrapositive are equivalent. This is a major proof tool because sometimes the contrapositive is easier to prove. Learning objective: Match a conditional with its contrapositive. 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 (Equivalent Conditional Statements): Which statement is equivalent to ¬(p ∧ q)?

  1. Worked Example: First identify exactly what the question is asking: Example case A (Equivalent Conditional Statements): Which statement is equivalent to ¬(p ∧ q)?
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. Use De Morgan's Law.
  4. Negating an and changes it to or.

Answer: ¬p ∨ ¬q

Practice problems

1. Practice case A (Equivalent Conditional Statements): Which statement is equivalent to ¬(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 (Equivalent Conditional Statements): Which statement is equivalent to ¬(p ∧ q)?
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. Use De Morgan's Law.
  4. Negating an and changes it to or.
  5. Negate both parts: ¬p ∨ ¬q.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: ¬p ∨ ¬q

2. Practice case B (Equivalent Conditional Statements): Which statement is equivalent to ¬(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 (Equivalent Conditional Statements): Which statement is equivalent to ¬(p ∨ q)?
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. Use De Morgan's Law.
  4. Negating an or changes it to and.
  5. Negate both parts: ¬p ∧ ¬q.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: ¬p ∧ ¬q

3. Practice case C (Equivalent Conditional Statements): Which statement is equivalent to p → q?

Choices: ¬q → ¬p · q → p · ¬p → ¬q · p ↔ q

Show solution
  1. Warm-up: First identify exactly what the question is asking: Practice case C (Equivalent Conditional Statements): Which statement is equivalent to p → q?
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. A conditional matches its contrapositive.
  4. Switch and negate both parts.
  5. That gives ¬q → ¬p.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: ¬q → ¬p

4. Practice case D (Equivalent Conditional Statements): How can a truth table show two statements are equivalent?

Choices: Their final columns match in every row. · They use the same number of letters. · One statement is longer. · Both contain an arrow.

Show solution
  1. Equivalence means same truth value in every case.
  2. Truth tables list every case.
  3. Matching final columns prove equivalence.

Answer: Their final columns match in every row.

5. Practice case E (Equivalent Conditional Statements): Simplify ¬¬q.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Practice case E (Equivalent Conditional Statements): Simplify ¬¬q.
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. Two negations cancel.
  4. The first negation flips q; the second flips back.
  5. So ¬¬q is q.
  6. Check the result by substituting or estimating: the response should match q and make sense in the original problem.

Answer: q

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