Proof by Deduction

Proof by deduction — also known as direct proof — is the most fundamental and widely used method of proof in A-Level Mathematics. It involves starting from known facts, definitions, axioms, or previously established results, and applying logical reasoning step by step until you reach the conclusion you wish to establish. Every step in a deductive proof must follow logically from the previous one.

The AQA A-Level Mathematics specification (7357) requires students to "construct and present mathematical arguments through appropriate use of diagrams; sketching graphs; logical deduction; precise statements involving correct use of symbols and connecting language." Proof by deduction is the backbone of this requirement.

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The Structure of a Deductive Proof

A deductive proof typically follows this pattern:

1. State clearly what you are trying to prove.
2. Introduce variables and define them precisely.
3. Perform a sequence of logical algebraic steps, each justified by known results.
4. Arrive at the required conclusion.
5. Write a clear concluding statement.

Exam Tip: Always define your variables. If you write "Let n be an integer," the examiner knows you understand what you are working with. Never assume the reader knows what your variables represent.

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Algebraic Representations for Proof

Before you can construct a deductive proof about properties of integers, you must be able to represent those properties algebraically. The following representations are essential:

Property	Algebraic Form
An even number	2n, where n is an integer
An odd number	2n + 1, where n is an integer
Consecutive integers	n, n + 1, n + 2, ...
Consecutive even numbers	2n, 2n + 2, 2n + 4, ...
Consecutive odd numbers	2n + 1, 2n + 3, 2n + 5, ...
A multiple of k	kn, where n is an integer
A number that leaves remainder r when divided by k	kn + r
A perfect square	n², where n is an integer

Important: When proving results involving two different integers, use different letters. For example, represent two even numbers as 2a and 2b, not 2n and 2n — the latter would force them to be the same number.

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Proving Results About Even and Odd Numbers

Example 1: Prove that the sum of two even numbers is always even.

Let the two even numbers be 2a and 2b, where a and b are integers.

2a + 2b = 2(a + b)

Since a + b is an integer (the integers are closed under addition), 2(a + b) is a multiple of 2, and therefore even.

Therefore, the sum of two even numbers is always even. ∎

Example 2: Prove that the sum of two odd numbers is always even.

Let the two odd numbers be 2a + 1 and 2b + 1, where a and b are integers.

(2a + 1) + (2b + 1) = 2a + 2b + 2 = 2(a + b + 1)

Since a + b + 1 is an integer, 2(a + b + 1) is a multiple of 2, and therefore even.

Therefore, the sum of two odd numbers is always even. ∎

Example 3: Prove that the product of two odd numbers is always odd.

Let the two odd numbers be 2a + 1 and 2b + 1, where a and b are integers.

(2a + 1)(2b + 1) = 4ab + 2a + 2b + 1 = 2(2ab + a + b) + 1

Since 2ab + a + b is an integer, the product has the form 2k + 1 for integer k, which is odd.

Therefore, the product of two odd numbers is always odd. ∎

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Proving Results About Divisibility

Example 4: Prove that for any integer n, n³ − n is divisible by 6.

n³ − n = n(n² − 1) = n(n − 1)(n + 1) = (n − 1)n(n + 1)

This is the product of three consecutive integers. Among any three consecutive integers:

• At least one is divisible by 2 (since every other integer is even).
• At least one is divisible by 3 (since every third integer is a multiple of 3).

Therefore, the product is divisible by both 2 and 3. Since 2 and 3 are coprime, the product is divisible by 2 × 3 = 6.

Therefore, n³ − n is divisible by 6 for all integers n. ∎

Example 5: Prove that for any integer n, n² + n is always even.

n² + n = n(n + 1)

This is the product of two consecutive integers. Of any two consecutive integers, exactly one is even. Therefore the product contains a factor of 2 and is even.

Therefore, n² + n is always even. ∎

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Proving Algebraic Identities

Example 6: Prove that (a + b)² − (a − b)² ≡ 4ab.

LHS = (a + b)² − (a − b)²
    = (a² + 2ab + b²) − (a² − 2ab + b²)
    = a² + 2ab + b² − a² + 2ab − b²
    = 4ab
    = RHS ∎

Example 7: Prove that (n + 3)² − (n + 1)² = 4(n + 2).

LHS = (n + 3)² − (n + 1)²
    = (n² + 6n + 9) − (n² + 2n + 1)
    = 4n + 8
    = 4(n + 2)
    = RHS ∎

Exam Tip: When proving an identity, always start from one side (usually the LHS) and manipulate it until you reach the other side. Never work from both sides simultaneously towards a middle expression — this is not logically valid as a proof.

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Proving Results About Squares

Example 8: Prove that if n is even, then n² is even.

Let n = 2k, where k is an integer.

n² = (2k)² = 4k² = 2(2k²)

Since 2k² is an integer, n² = 2(2k²) is even.

Therefore, if n is even, then n² is even. ∎

Example 9: Prove that if n is odd, then n² is odd.

Let n = 2k + 1, where k is an integer.

n² = (2k + 1)² = 4k² + 4k + 1 = 2(2k² + 2k) + 1

Since 2k² + 2k is an integer, n² has the form 2m + 1, which is odd.

Therefore, if n is odd, then n² is odd. ∎

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Common Pitfalls in Deductive Proof

1. Using the same variable for different quantities. Writing "Let the two even numbers be 2n and 2n" forces them to be the same number.

2. Assuming what you are trying to prove. This is circular reasoning. You must start from known facts and derive the conclusion.

3. Checking specific cases instead of proving generally. Showing that a result works for n = 1, 2, 3 is not a proof — it is verification of particular cases.

4. Missing the concluding statement. Always write a sentence at the end that directly states what has been proved.

5. Not defining variables. Always state "Let n be an integer" or similar before using a variable.

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Summary

• Proof by deduction (direct proof) uses logical steps from known facts to reach a conclusion.
• Represent even numbers as 2n, odd numbers as 2n + 1, and use different letters for different quantities.
• For divisibility proofs, factorise the expression and identify factors.
• For identity proofs, work from one side to the other.
• Always define your variables, show every algebraic step, and write a clear concluding statement.
• Three consecutive integers always include a multiple of 2 and a multiple of 3, so their product is divisible by 6.

Exam Tip: In AQA exam papers, proof by deduction questions often ask you to "Prove that..." followed by an algebraic statement. Read the question carefully — if it says "for all integers n," you must give a general proof, not check specific values. Structure your proof clearly: define variables, perform algebraic manipulation, and conclude with a sentence that refers back to the original statement.