Simple deductive proof

Mathematics Analysis and Approaches SL· Number & Algebra, Section 1.7· 12 min read

1. Core Principles of Deductive Reasoning★★☆☆☆⏱ 3 min

Unlike inductive reasoning that uses specific examples to guess a general rule, deductive reasoning starts from universal definitions to produce a conclusion that is 100% guaranteed true for all cases in the set. This is the standard of rigor required for all IB AA SL proof questions.

📘 Definition

Even Integer

2k

Any integer that can be written as 2 multiplied by some other integer k

Example:

4 = 2(2), -6 = 2(-3)

📐 Worked Example

Prove that the sum of any two even integers is even

1
Define two arbitrary even integers using the formal definition
2
$Let a = 2m, b = 2n \text{ where } m,n \in \mathbb{Z}$
3
Calculate their sum and factor out the common 2 term
4
$a + b = 2m + 2n = 2(m + n)$
5
Since the sum of two integers m + n is also an integer, the total sum matches the definition of an even integer, so the statement holds for all even integers.

✓ Quick check

Test your understanding of general vs specific cases

Which of the following is a valid deductive starting point for a proof about all odd integers?
- Let x = 3, which is odd
- Let x = 2k + 1 where k is any integer
- Let x be any positive odd number less than 10
Reveal answer
Let x = 2k + 1 where k is any integer —
This definition applies to every odd integer across the full set, no exceptions.

Exam tip:

Examiners will deduct 1-2 marks automatically if you do not explicitly define your variables as members of the integer set at the start of your proof.

2. Parity Property Proofs★★☆☆☆⏱ 3 min

Parity proofs are the most common introductory deductive proof question, requiring you to prove properties of sums, differences, and products of even and odd numbers. All parity rules follow directly from the 2k and 2k+1 definitions.

📐 Worked Example

Prove that the product of any two odd integers is always odd

1
Define two arbitrary odd integers
2
$Let p = 2a + 1, q = 2b + 1 \text{ where } a,b \in \mathbb{Z}$
3
Expand their product fully
4
$pq = (2a+1)(2b+1) = 4ab + 2a + 2b + 1$
5
Factor out 2 from all terms except the final 1
6
$pq = 2(2ab + a + b) + 1$
7
Since 2ab + a + b is an integer, the expression matches the 2k+1 definition of an odd number, so the product is always odd.

3. Simple Divisibility Proofs★★★☆☆⏱ 3 min

Divisibility proofs use the formal definition that if d divides integer n, then n = dk for some integer k. You will be expected to expand and factor algebraic expressions to show this structure holds.

📐 Worked Example

Prove that for all positive integers n, 3 divides n³ + 2n

1
Rewrite the expression to factor or rearrange terms
2
$n^3 + 2n = n(n^2 + 2)$
3
Consider the three possible cases for n modulo 3: n=3k, n=3k+1, n=3k+2
4
Case 1: If n = 3k, the expression becomes 3k(9k² + 2), which is clearly divisible by 3
5
Case 2: If n = 3k+1, n² + 2 = 9k² +6k +1 +2 = 3(3k² + 2k +1), so the product is divisible by 3
6
Case 3: If n = 3k+2, n² + 2 =9k² +12k +4 +2 = 3(3k² +4k +2), so the product is divisible by 3
7
Since all three possible cases for n give a result divisible by 3, the statement holds for all positive integers.

4. Disproving Conjectures with Counterexamples★★☆☆☆⏱ 3 min

You do not need a full deductive proof to disprove a universal statement. A single valid counterexample that violates the conjecture is sufficient to earn full marks for this type of exam question.

5. Common Pitfalls

Wrong move:

Using specific test cases to "prove" a general universal statement

Why:

Inductive reasoning from a small number of examples does not guarantee the statement holds for all cases, and will not earn full marks

Correct move:

Start with general variable definitions that apply to every member of the set you are working with

Wrong move:

Assuming the conclusion you are supposed to prove at the start of your argument

Why:

This creates circular reasoning that is logically invalid, and examiners will award zero marks for the full proof

Correct move:

Only start from agreed definitions, axioms, and results you have already proven earlier in the question

Wrong move:

Forgetting to explicitly define variables as integers before using parity properties

Why:

Parity only applies to whole numbers, so unstated assumptions are considered a logical gap by IB examiners

Correct move:

Write "let n be any integer" immediately before introducing parity rules for your variable

Wrong move:

Presenting a counterexample as proof of a true universal statement

Why:

A counterexample can only disprove a conjecture, it can never be used to prove a statement is true for all cases

Correct move:

Reserve counterexamples exclusively for disproving false claims, use deductive steps for true universal statements

Wrong move:

Skipping intermediate algebraic simplification steps in your working

Why:

Examiners cannot award method marks if they cannot follow the full logical flow of your argument

Correct move:

Write every simplification step clearly, no matter how trivial it seems to you

6. Quick Reference Cheatsheet

Proof Type	Starting Point	Required Final Step	Common Use Case
Direct deductive proof	General definition of the full set	Unambiguous confirmation conclusion holds for all members	Parity, divisibility, algebraic identities
Disproof by counterexample	Universal false conjecture	Single valid example violating the conjecture	Showing a statement is not universally true
Proof by exhaustion	Finite predefined set	Verify every member of the set meets the conclusion	Small finite number ranges, e.g. 1 to 10

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2023 · Paper 1
Simple divisibility proof
2022 · Paper 2
Parity property proof
2021 · Paper 1
Algebraic identity proof

What's Next

Mastering simple deductive proof builds the foundational logical framework for all further proof work. In AA SL you will keep applying these deductive and disproof-by-counterexample techniques across the course; if you continue to AA HL, they extend to proof by mathematical induction and proof by contradiction. These deductive reasoning skills also transfer directly to your IA exploration, where you will need to justify general mathematical claims rather than relying on specific test cases. Before moving on, ensure you can independently construct full proofs for the sum of two odd integers, divisibility of 4 for n² -1 for odd n, and the difference of squares identity. These skills will help you avoid costly logical errors that often separate 6 and 7 grade responses.