Proof and Reasoning | IB Math AAHL | Perplex

Proof and Reasoning

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Watch comprehensive video reviews for Proof and Reasoning, designed for final exam preparation. Each video includes integrated problems you can solve alongside detailed solutions.

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Interactive Problems: Solve problems alongside the video with step-by-step guidance and detailed solutions.

Exam Preparation: Complete unit reviews designed for final exam preparation with all key concepts covered systematically.

Expert Teaching: High-quality instruction from Perplex co-founder James Mullen with clear explanations, worked examples, and exam tips.

Not your average video:

Interactive Problems: Solve problems alongside the video with step-by-step guidance and detailed solutions.

Exam Preparation: Complete unit reviews designed for final exam preparation with all key concepts covered systematically.

Expert Teaching: High-quality instruction from Perplex co-founder James Mullen with clear explanations, worked examples, and exam tips.

Direct proof (SL + HL)HL Terms & DefinitionsContradiction (HL)Proof by induction

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Proof by contradiction

AHL AA 1.15

To prove a statement by contradiction, we essentially show that if the statement were not true, math would "break".

In practice, we assume the opposite of the statement, and and derive a contradiction - a mathematical inconsistency.

Example

Prove that there is no smallest positive real number.

Suppose that there exists a smallest positive real number and call it r.

Since r is real and positive, 2r is also real and positive.

But 2r<r, contradicting the assumption that it is the smallest positive real number.

Hence, there is no smallest positive real number.

Note that they "key" is that this argument works for any r. No matter what positive real number someone suggests, we know that half that number is also a positive real number.

Proof by contradiction

AHL AA 1.15

To prove a statement by contradiction, we essentially show that if the statement were not true, math would "break".

In practice, we assume the opposite of the statement, and and derive a contradiction - a mathematical inconsistency.

Example

Prove that there is no smallest positive real number.

Suppose that there exists a smallest positive real number and call it r.

Since r is real and positive, 2r is also real and positive.

But 2r<r, contradicting the assumption that it is the smallest positive real number.

Hence, there is no smallest positive real number.

Note that they "key" is that this argument works for any r. No matter what positive real number someone suggests, we know that half that number is also a positive real number.