Proof and Reasoning | IB Math AAHL | Perplex

Proof and Reasoning

Video Reviews

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

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.

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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Deductive Proof

SL AA 1.6

A proof is a strict logical argument that demonstrates with mathematical certainty that a statement is true.

For SL students, these proofs will be in the form

Prove that

(LHS expression)≡(RHS expression)

where ≡ means equivalent, ie equal for ALL variables in the expression, not just some specific intersections.

We use LHS (left-hand side) and RHS (right-hand side) as an abbreviation for one side of the equivalence.

Example

Prove that

√a+√b1≡a−b√a−√b,0<b<a

Starting with the LHS and rationalizing the denominator:

√a+√b1≡√a+√b1⋅√a−√b√a−√b≡a−b√a−√b

Q.E.D.

The abbreviation Q.E.D (quod erat demonstrandum - latin for "that's what we wanted to prove") is a fancy way of terminating a proof.

Notice that the key step here - rationalizing the denominator, does not change the LHS, since we are essentially multiplying by 1. This is the key to all deductive proofs - making algebraic changes that do not actually change the expression fundamentally.

It's very important when laying out proofs to pick one side, and perform steps to show equivalence with the other side. You should never manipulate both sides at once, as this can lead to logical errors and lose you marks.

Deductive Proof

SL AA 1.6

A proof is a strict logical argument that demonstrates with mathematical certainty that a statement is true.

For SL students, these proofs will be in the form

Prove that

(LHS expression)≡(RHS expression)

where ≡ means equivalent, ie equal for ALL variables in the expression, not just some specific intersections.

We use LHS (left-hand side) and RHS (right-hand side) as an abbreviation for one side of the equivalence.

Example

Prove that

√a+√b1≡a−b√a−√b,0<b<a

Starting with the LHS and rationalizing the denominator:

√a+√b1≡√a+√b1⋅√a−√b√a−√b≡a−b√a−√b

Q.E.D.

The abbreviation Q.E.D (quod erat demonstrandum - latin for "that's what we wanted to prove") is a fancy way of terminating a proof.

Notice that the key step here - rationalizing the denominator, does not change the LHS, since we are essentially multiplying by 1. This is the key to all deductive proofs - making algebraic changes that do not actually change the expression fundamentally.

It's very important when laying out proofs to pick one side, and perform steps to show equivalence with the other side. You should never manipulate both sides at once, as this can lead to logical errors and lose you marks.