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SL 1.6 · Use simple deductive proof and disprove statements with counterexamples

Learn to use simple deductive proof and disprove statements with counterexamples through clear examples and targeted practice.

International Baccalaureate (IB) IB AA SL: Mathematics: Analysis and Approaches SL

Number and Algebra

How to justify a statement—and how one example can show it is false

Mathematics often asks whether a statement is true for every value in a stated set. Testing a few values can help you notice a pattern, but it cannot establish that the pattern always holds. A proof gives a chain of reasons showing that a conclusion follows from accepted facts and definitions. To show that a universal statement is false, however, just one valid counterexample is enough. This lesson uses familiar algebra and simple numerical or graphical checks; it does not rely on advanced proof methods.

What you will learn

Explain the difference between checking examples and proving a general statement.
Use definitions and familiar algebra to build a simple deductive proof.
Disprove a universal statement by giving a valid counterexample.
Use numerical or graphing technology as a check without treating it as proof.

1. Prior knowledge: statements, examples, and scope

A mathematical statement is a sentence that can be judged true or false. Pay close attention to words such as “all,” “every,” and “for any.” A statement that says a property holds for every number in a set is called a universal statement. Its domain—the set of values being considered—is part of its meaning. For example, a claim about every integer is not the same as a claim about every real number.

A proof must respect the domain and any conditions in the statement. If a claim concerns even integers, a test value must be an even integer. An example outside the stated domain cannot confirm or disprove the claim. Recall that an even integer can be written as

2k

for some integer

k

; this definition will let us reason about all even integers at once.

A numerical test checks only the values you choose. Several successful tests may suggest that a statement is true, but there may be an untested value for which it fails. A counterexample is a specific value in the stated domain that makes the claim false.

Read the domain and conditions before testing a statement.
Examples can suggest a rule; they do not prove a universal rule.
One valid counterexample disproves a universal statement.

2. Simple deductive proof: make the reasoning general

A deductive proof begins with the assumptions in the statement and uses definitions or algebraic facts to reach the conclusion. Each step must follow from the previous information. For example, instead of checking a few even numbers, represent an arbitrary even integer as

2k

, where

k

is an integer. Because

k

is not a chosen example, the reasoning applies to every even integer.

A useful structure is: state what is assumed, express that assumption using a definition, transform the expression carefully, and identify why the final expression has the required property. For an algebraic identity, start with one side and use valid algebra to obtain the other. Do not use the identity you are trying to prove as a reason for a step.

The word “arbitrary” means that the value is not specially chosen beyond meeting the stated conditions. In a proof, let an arbitrary permitted value stand for all values in the domain. The proof works only if no extra condition is quietly imposed on that value.

Translate definitions into algebraic form.
Show a connected chain of justified steps.
A proof must cover every value allowed by the statement.

3. Counterexamples and representations

To disprove a universal statement, find one permitted input for which the conclusion fails. Substitute it into the claim and show the mismatch. If the claim concerns all integers, use an integer; if it concerns positive numbers, use a positive number. A value that violates a condition is not a counterexample.

Numbers, algebra, graphs, and contexts can all help you investigate a claim. A table of several input-output values may reveal a possible pattern, while algebra can explain why it holds generally. A graph may show that two expressions appear different at some point, but a screen displays only a finite picture and may hide small differences or behaviour outside its viewing window.

Graphing technology is therefore useful for exploring and checking. For example, graph two proposed expressions on the same axes or make a table of values to look for a likely counterexample. Then give the mathematical reason: an exact substitution for a disproof, or a chain of algebraic reasoning for a proof. Calculator output by itself is not a deductive proof.

For a universal claim, seek one in-domain value where it fails.
A numerical or graphical check is evidence for exploration, not a complete proof.
State exact values where possible so the counterexample is unambiguous.

4. Exam-style communication

In a written response, make your purpose clear. If asked to prove a statement, explain why your reasoning covers all allowed values. If asked to disprove it, name the counterexample and show that it satisfies the conditions but not the conclusion. A bare answer such as “false” or a list of tested values is not enough.

Check the logic as well as the arithmetic. In a proof, each transformation must preserve equality or follow from a definition. In a disproof, verify both sides of the task: the chosen value belongs to the domain, and it makes the claimed conclusion fail.

Use sentences to explain what each algebraic step establishes.
For a disproof, explicitly verify domain membership and failure.
Do not claim a statement is true for all values based only on a graph or sample.

Worked example

Prove a statement about even integers

Prove that the sum of any two even integers is even.

Represent the assumptions
Let the two even integers be arbitrary. By the definition of an even integer, each is twice an integer; use separate integer variables for them.
$a=2m, b=2n, m,n∈\mathbb{Z}$
Add and factor
Add the expressions for the two integers and factor out $2$ . This is valid algebra and expresses the sum as twice another number.
$a+b=2m+2n=2(m+n)$
Apply the definition
Since integers are closed under addition, $m+n$ is an integer. Therefore the sum is twice an integer, which is exactly the definition of an even integer.
$m+n∈\mathbb{Z}$

Answer: The sum of any two even integers is even.

Check: The proof uses arbitrary even integers, not just selected examples, so it covers every pair in the stated domain.

Worked example

Disprove a claim about squares

Disprove the statement: for every integer

n

, the number

n^2+n+1

is even.

Choose a permitted value
A single integer that makes the claim fail is enough. Choose $n=1$ , which belongs to the stated domain of integers.
$n=1∈\mathbb{Z}$
Evaluate the expression
Substitute the chosen value into the expression. The result is $3$ , which is odd rather than even.
$1^2+1+1=3$

Answer: The statement is false;

n=1

is a counterexample.

Check: The selected input is an integer and its output is not even, so it satisfies the counterexample requirements.

Worked example

Use algebra to prove an identity

Prove that

(x+3)^2-(x-3)^2=12x

for every real number

x

Expand both squares
Use the square rule $(u+v)^2=u^2+2uv+v^2$ and its corresponding form with subtraction. These expansions are valid for every real $x$ .
$(x+3)^2-(x-3)^2=(x^2+6x+9)-(x^2-6x+9)$
Simplify
Distribute the subtraction across the second bracket. The $x^2$ terms and constant terms cancel, leaving the stated expression.
$x^2+6x+9-x^2+6x-9=12x$
Connect to a technology check
A graphing calculator can display the two sides as overlapping graphs, or a table can compare their values for selected inputs. This is a useful check, but the algebra above proves equality for every real $x$ .

Answer: The identity holds for all real numbers

x

Check: The reasoning uses valid expansions and simplification without restricting

x

, so the conclusion applies throughout the stated real-number domain.

Common mistakes and how to avoid them

Checking several values and concluding that a statement is true for every value.

Correction: Use the checks to look for patterns, then prove the statement with reasoning that applies to an arbitrary value.

Giving a value that does not belong to the stated domain as a counterexample.

Correction: Check the conditions first; the counterexample must be one of the values covered by the claim.

Writing a counterexample value without showing what happens to the claim.

Correction: Substitute the value and state explicitly why the result contradicts the conclusion.

Treating a graph or calculator display as proof.

Correction: Use technology to explore or check, then justify the result with exact substitution or algebra.

Lesson summary

A simple deductive proof uses assumptions, definitions, and valid steps to establish a conclusion for all values in a domain.
Testing examples can suggest a pattern but cannot prove a universal claim.
One valid counterexample is enough to disprove a universal statement.
Numerical and graphical technology can support exploration, but mathematical reasoning must justify the conclusion.

Check your understanding

Question 1

Which is enough to disprove the statement “every integer greater than

1

is even”?

Check that $4$ is even.
Use $3$ , since it is an integer greater than $1$ and is odd.
Use $1$ , since it is odd.
Graph the even integers.

Show answer and explanation

Use

3

, since it is an integer greater than

1

and is odd.

3

meets the condition and fails the claimed conclusion, so it is a valid counterexample. The value

1

is outside the stated condition.

Question 2

What is the key feature of a proof that the sum of two even integers is even?

It checks many even-number pairs.
It represents arbitrary even integers as twice integers and shows their sum has the same form.
It uses a graph to show the sum is always positive.
It chooses one pair whose sum is even.

Show answer and explanation

It represents arbitrary even integers as twice integers and shows their sum has the same form.

Writing arbitrary even integers as

2m

and

2n

lets the algebra cover every permitted pair.

Question 3

A calculator table gives matching values for two expressions at ten inputs. What conclusion is justified by the table alone?

The expressions are equal for every real input.
The expressions are equal only at the ten displayed inputs.
The expressions are equal for all integers.
The expressions cannot be compared.

Show answer and explanation

The expressions are equal only at the ten displayed inputs.

The table confirms the comparison only at the displayed inputs. A general claim needs a proof or, if false, a counterexample.

Key terms

Universal statement: A claim that a property holds for every value in a specified domain.
Domain: The set of values that a statement allows.
Deductive proof: A connected argument using accepted facts and valid reasoning to establish a conclusion.
Counterexample: A value in the stated domain that shows a universal claim is false.
Arbitrary: Not specially chosen, except for meeting the conditions in the statement.

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Published by DoAssignment. This AI-assisted lesson follows International Baccalaureate (IB) IB AA SL: Mathematics: Analysis and Approaches SL, study topic SL 1.6. It is a study resource, not an official curriculum publication.

Before publication, the draft is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability, and then requires administrator approval. Errors can still occur, so corrections are welcomed.

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