Analyze and prove conjectures using inductive and deductive reasoning

Alberta Grade 11 Mathematics

Number and Logic

How to explore a conjecture, test it, and decide whether it is true

In mathematics, a conjecture is a statement that seems true but has not yet been proved. You can look for patterns in examples to form a conjecture. That is inductive reasoning. You can also use facts and logical steps to show that a statement must be true. That is deductive reasoning. This lesson shows how examples can suggest an idea, but why a general argument is needed to prove it.

What you will learn

Distinguish inductive reasoning from deductive reasoning.
Use examples to form and test a conjecture.
Use a general argument to prove a conjecture.
Use a counterexample to show that a conjecture is false.

1. Review: examples, variables, and statements

An example is one specific case. For instance, 4 is one even number. A variable is a letter that stands for a number, such as

n

. A statement makes a claim that can be true or false.

An even integer is an integer that can be written as

2k

for some integer

k

. An odd integer can be written as

2k+1

. These forms let us describe all even or odd integers, not just a few examples.

A conjecture is a statement based on observations or reasoning that still needs to be checked. A proof is a connected argument that shows a statement is true in every case covered by the statement.

A few examples can suggest a pattern, but they do not cover every possible case.
Variables help describe a whole group of numbers.

2. Inductive reasoning: observe, test, and conjecture

Inductive reasoning starts with specific cases. You look for a pattern and use it to suggest a general statement. The general statement is a conjecture, not yet a proof.

Suppose you add the first few odd numbers. The results may suggest that the sum of the first

n

odd numbers is a square number. Checking more cases can make the idea seem convincing, but any finite list of checks leaves other cases unchecked.

A useful test is to try small cases, unusual cases, and cases that might challenge the claim. If one case fails, the conjecture as stated is false. If every tested case works, the conjecture may be worth proving, but it is not proved just by testing.

Inductive reasoning moves from specific observations to a possible general claim.
Examples support a conjecture but do not prove a claim about all cases.
One counterexample is enough to disprove a claim that is supposed to be true in every case.

3. Deductive reasoning: prove or disprove

Deductive reasoning uses accepted facts and valid steps to reach a conclusion. To prove a conjecture, start with what is given and explain why each step follows. The argument must cover every case in the claim.

For example, to prove a claim about all even integers, represent an arbitrary even integer as

2k

. The word arbitrary means that

k

is not a chosen special value; it can be any integer. If the reasoning works for this general form, it applies to every even integer.

A direct proof begins with the conditions in the claim and works toward its conclusion. A counterexample is one specific case that satisfies the claim's conditions but fails its conclusion. It disproves a universal claim without needing to check every case.

A direct proof links the given condition to the conclusion with valid steps.
Use an arbitrary variable to represent all values in the stated group.
A counterexample must meet the original conditions.

4. A practical approach

First, state the claim clearly. Identify which values it covers and what it says about them. Next, test examples to see whether the claim is plausible. If a test fails, explain why it is a counterexample.

If the examples support the claim, look for a general reason. Use definitions or known facts, and connect each step to the next. At the end, check that the argument addresses the full claim rather than only the examples you tried.

For a student try, consider the statement: the sum of two odd integers is odd. Test a few pairs, then decide whether a general argument can prove it. Do not treat the tests alone as proof.

Separate testing a claim from proving it.
Check that a proof covers every value described in the claim.
Read the conclusion again to make sure the argument actually establishes it.

Worked example

From a pattern to a conjecture

Find a pattern in the sums of the first odd numbers. State a conjecture, and explain whether the examples prove it.

Calculate cases
Add the first one, two, and three odd numbers. Each result is a square number.
$1=1^2,\quad 1+3=4=2^2,\quad 1+3+5=9=3^2$
State the conjecture
The pattern suggests that adding the first $n$ positive odd integers gives $n^2$ . The examples support this conjecture, but checking three values does not prove it for every positive integer $n$ .
$1+3+\cdots +(2n-1)=n^2$

Answer: The examples suggest that the sum of the first

n

positive odd integers is

n^2

. This is an inductive conjecture, not a proof.

Check: The claim concerns every positive integer

n

, while the calculations check only three cases.

Worked example

A direct proof about even integers

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

Represent the integers
By the definition of an even integer, write the two even integers as $2a$ and $2b$ , where $a$ and $b$ are integers.
$2a+2b$
Factor the sum
Factor out 2. Since the sum of two integers is an integer, $a+b$ is an integer. Therefore the total has the form 2 times an integer, which is the definition of even.
$2a+2b=2(a+b)$

Answer: The sum is even because it can be written as 2 times the integer

a+b

. This argument applies to any two even integers.

Check: For example, 8 and 14 are even, and their sum 22 is even. This check agrees with the proof but is not a replacement for it.

Worked example

Disproving a claim with a counterexample

A student claims that the square of every odd integer is even. Decide whether the claim is true.

Test an odd integer
Choose 3. It is odd, so it meets the condition in the claim.
$3^2=9$
Compare with the conclusion
The result 9 is odd, not even. This one case satisfies the condition but fails the conclusion, so it is a counterexample.
$9=2(4)+1$

Answer: The claim is false. The odd integer 3 has an odd square, 9, so it is a counterexample.

Check: A counterexample needs only one valid case to disprove a claim about every odd integer.

Common mistakes and how to avoid them

Calling a pattern proved after checking several examples.

Correction: Call it a conjecture until a general argument establishes it for all cases.

Using a number that does not meet the claim's condition as a counterexample.

Correction: Check that the chosen case satisfies the condition before showing that it fails the conclusion.

Using a variable as though it were one particular number.

Correction: State what the variable represents and show that the argument works for every value allowed.

Lesson summary

Inductive reasoning uses examples to notice patterns and form conjectures.
A conjecture needs a general argument before it is proved.
Deductive reasoning uses definitions and valid steps to establish a claim.
A counterexample disproves a claim that is meant to hold in every case.

Check your understanding

Question 1

A student checks four cases of a claim about all integers, and all four work. What has the student established?

The claim is proved for all integers.
The examples support the claim, but it is not yet proved.
The claim is false.
The claim applies only to those four integers.

Show answer and explanation

The examples support the claim, but it is not yet proved.

A finite number of examples can support a conjecture but cannot establish a claim about all integers.

Question 2

Which statement proves that the sum of two odd integers is even?

The sum of 3 and 5 is 8.
Every odd integer is positive.
Write the integers as $2a+1$ and $2b+1$ ; their sum is $2(a+b+1)$ , which is even.
The sum is even because odd numbers are close to even numbers.

Show answer and explanation

Write the integers as

2a+1

and

2b+1

; their sum is

2(a+b+1)

, which is even.

The expression represents any two odd integers. The sum is 2 times an integer, so the proof covers every pair.

Question 3

A claim says every multiple of 4 is even. Which is the best way to verify it deductively?

Check 4, 8, and 12.
Represent any multiple of 4 as $4k=2(2k)$ ; since $2k$ is an integer, it is even.
Find a multiple of 4 that is odd.
Check only the largest multiple you can think of.

Show answer and explanation

Represent any multiple of 4 as

4k=2(2k)

; since

2k

is an integer, it is even.

This uses the general form of a multiple of 4 and shows it is twice an integer, which is the definition of even.

Key terms

Conjecture: A statement that seems true based on observations or reasoning but has not yet been proved.
Inductive reasoning: Reasoning from specific examples to a possible general statement.
Deductive reasoning: Reasoning from accepted facts through valid steps to reach a conclusion.
Proof: A connected argument that establishes a statement for every case it covers.
Counterexample: A case that meets a claim's condition but shows that its conclusion fails.

Continue through Math 20-2

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Alberta Grade 11 Mathematics (Math 20-2), expectation NL1. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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