Proof by Counterexample
A free Logic lesson from the “Proof Readiness” unit, with a worked example and practice problems including step-by-step solutions.
A proof by counterexample does not need many cases. One precise case can disprove a universal claim if it meets the setup and fails the conclusion. Learning objective: Disprove all claims with a single correct counterexample. 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: A direct explanation starts with definitions and moves forward to the conclusion. Example 2: A counterexample must satisfy the setup and break the conclusion. 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
- Disprove all claims with a single correct counterexample
- Explain the idea in plain English before using symbols
- Use examples, non-examples, or counterexamples to check the reasoning
Worked example
Problem. Example case A (Proof by Counterexample): Worked example: Which rule or habit best matches Proof by Counterexample?
- Proof by Counterexample targets a specific reasoning habit.
- Proof by Counterexample focuses on disprove all claims with a single correct counterexample.
- The other choices either overclaim or change the logical relationship.
Answer: Proof by Counterexample focuses on disprove all claims with a single correct counterexample.
Practice problems
1. Practice case A (Proof by Counterexample): Which explanation is most complete?
Choices: Because x is even, x = 2k for an integer k, so x + 2 = 2(k + 1), which is even. · It stays even. · I tried x = 4. · The answer looks right.
Show solution
- Warm-up: First identify exactly what the question is asking: Practice case A (Proof by Counterexample): Which explanation is most complete?
- For a counterexample, find one case that satisfies the setup but makes the conclusion false.
- A complete explanation uses a definition.
- It shows the algebraic step.
- It connects the result back to evenness.
- Verify the selected choice by checking that it preserves the stated logical meaning and that the other choices change the rule or claim.
Answer: Because x is even, x = 2k for an integer k, so x + 2 = 2(k + 1), which is even.
2. Practice case B (Proof by Counterexample): Direct reasoning usually starts with:
Choices: the given information and definitions · the opposite of the conclusion · a random answer choice · a diagram with no labels
Show solution
- Warm-up: First identify exactly what the question is asking: Practice case B (Proof by Counterexample): Direct reasoning usually starts with:
- For a counterexample, find one case that satisfies the setup but makes the conclusion false.
- Direct reasoning moves forward.
- It begins from what is given.
- Definitions and known facts justify each step.
- Verify the selected choice by checking that it preserves the stated logical meaning and that the other choices change the rule or claim.
Answer: the given information and definitions
3. Practice case C (Proof by Counterexample): To disprove "All multiples of 4 are multiples of 8," which counterexample works?
Choices: 4 · 8 · 16 · 24
Show solution
- Warm-up: First identify exactly what the question is asking: Practice case C (Proof by Counterexample): To disprove "All multiples of 4 are multiples of 8," which counterexample works?
- For quantified statements, identify the domain first, then decide whether the claim is about all objects or at least one object.
- A counterexample must be a multiple of 4.
- 4 is not a multiple of 8.
- So 4 disproves the claim.
- Verify the selected choice by checking that it preserves the stated logical meaning and that the other choices change the rule or claim.
Answer: 4
4. Practice case D (Proof by Counterexample): Reasoning by cases is appropriate when:
Choices: the cases cover all possibilities · only one example is checked · the conclusion is ignored · the domain is unknown
Show solution
- Casework splits a problem into possibilities.
- The proof is complete only if every possibility is covered.
- Then each case can be handled separately.
Answer: the cases cover all possibilities
5. Practice case E (Proof by Counterexample): Lesson focus: Which transfer task would show readiness for Proof by Counterexample?
Choices: Apply disprove all claims with a single correct counterexample in a new sentence or context. · Repeat the exact worked example without changing the context. · Pick an answer before identifying the claim parts. · Use a rule from another lesson because the words sound close.
Show solution
- A transfer task keeps the same objective but changes the surface context.
- Apply disprove all claims with a single correct counterexample in a new sentence or context.
- That is a better mastery signal than memorizing a practice prompt.
Answer: Apply disprove all claims with a single correct counterexample in a new sentence or context.
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