Lesson 2.1.2

Finding Counterexamples

One single example that breaks a conjecture is all it takes to disprove it — that powerful example is called a counterexample.

Introduction

You can support a conjecture with a million examples, but it only takes one counterexample to destroy it. Learning to find counterexamples trains you to think critically about every claim.

Past Knowledge

Inductive reasoning & conjectures (2.1.1).

Today's Goal

Identify and construct counterexamples to disprove false conjectures.

Future Success

Conditional statements (2.1.3) and proofs (2.3) build on critical-thinking skills.

Key Concepts

What Is a Counterexample?

A counterexample is a specific case for which a conjecture is false. Finding even one counterexample proves the conjecture is wrong.

Strategy for Finding Counterexamples

Read the conjecture carefully — what does it claim?
Think of edge cases: negative numbers, zero, obtuse angles, degenerate shapes.
Test boundary conditions and extremes.

One Is Enough

You only need one counterexample to disprove a conjecture — no matter how many examples supported it.

Worked Examples

Basic

Number Conjecture

Conjecture: “The square of any number is greater than the number itself.” True or false?

Try : .

Counterexample: . The conjecture is false.

Intermediate

Geometry Conjecture

Conjecture: “All supplementary angles are a linear pair.” True or false?

Two non-adjacent angles measuring 60° and 120° are supplementary but do not share a side — they are not a linear pair.

Counterexample: Non-adjacent angles of 60° and 120°. The conjecture is false.

Advanced

Coplanarity Conjecture

Conjecture: “Any four points are coplanar.” True or false?

Three corners of a table and the tip of a pen held above the table form a non-coplanar set.

Counterexample: Four vertices of a tetrahedron. The conjecture is false.

Common Pitfalls

Only Testing “Nice” Numbers

Counterexamples often hide in fractions, negatives, or zero. Always test edge cases beyond simple positive integers.

Real-Life Applications

Software Testing

Software testers are professional counterexample finders. They try edge cases, unusual inputs, and extreme conditions to find the one case that breaks the program — just like finding the one counterexample that disproves a conjecture.