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Quick questions on Proof: deduction, exhaustion, counter-example and contradiction - Edexcel A-Level Maths

5short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.

What is disproof by counter-example?

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To disprove a general statement you only need one example where it fails. The single example must be shown to break the claim explicitly.

What is inequality by deduction?

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Prove that

a^2 + b^2 \ge 2ab

for all real

a

and

b

. Start from the fact that a square is non-negative:

(a-b)^2 \ge 0

. Expanding gives

a^2 - 2ab + b^2 \ge 0

, so

a^2 + b^2 \ge 2ab

What is exhaustion on a small range?

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Prove that every integer

n

with

1 \le n \le 5

satisfies

n^2 \le 2^n + 1

. Check each:

n=1

gives

1 \le 3

;

n=2

gives

4 \le 5

;

n=3

gives

9 \le 9

;

n=4

gives

16 \le 17

;

n=5

gives

25 \le 33

. All five cases hold, so the statement is proved.

What is q1?

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Prove that the sum of two consecutive integers is always odd. [3 marks]

What is q2?

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Disprove the claim that

2^n + 1

is prime for all positive integers

n

. [2 marks]

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