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IBMathsDPAnalysis & Approaches (AA)SLRevision Notes1. Number & AlgebraProof & ReasoningProof

Proof (DP IB Analysis & Approaches (AA)): Revision Note

Amber

Written by: Amber

Reviewed by: Mark Curtis

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DP Analysis & Approaches (AA): SLDP Analysis & Approaches (AA): HLDP Analysis & Approaches (AA): HL

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Language of Proof

 What is proof?

  • Proof is a series of logical steps which show that a result (statement) is true for all specified numbers

    • It is not enough to prove a statement by testing just a few numbers

  • Algebra is commonly used to help prove statements

What notation do I need to know?

  • You need to be familiar with the following notation

    • L H S is left-hand side

    • R H S is right-hand side

    • identical to is the identity symbol

      • The LHS equals the RHS for all values of x

      • e.g. x plus x identical to 2 x (there's nothing to 'solve')

      • e.g. negative x open parentheses 1 minus x close parentheses identical to negative x plus x squared

    • straight integer numbers is the set of integers open curly brackets 0 comma space plus-or-minus 1 comma space plus-or-minus 2 comma space plus-or-minus 3 comma space... close curly brackets

      • straight integer numbers to the power of plus is the set of positive integers open curly brackets 1 comma space 2 comma space 3 comma space... close curly brackets

    • straight natural numbers is the set of natural numbers

      • open curly brackets 0 comma space 1 comma space 2 comma space 3 comma space... close curly brackets

      • Zero is included, unlike straight integer numbers to the power of plus

    • straight rational numbers is the set of rational numbers

      • These are numbers in the form a over b where a comma space b element of straight integer numbers and b not equal to 0

      • e.g. 2 over 3 comma space 8 open parentheses equals 8 over 1 close parentheses comma space minus 1 over 7 comma space 0 equals open parentheses 0 over 1 close parentheses comma space...

      • straight rational numbers to the power of plus is the set of positive rational numbers

    • straight real numbers is the set of real numbers

      • straight real numbers to the power of plus is the set of positive real numbers, open curly brackets x space vertical line space x element of straight real numbers comma space x greater than 0 close curly brackets

Diagram of number sets: natural numbers (N) inside integers (Z) inside rationals (Q) inside reals (R), each set nested within an oval shape.

Examiner Tips and Tricks

Whilst the identity symbol identical to is in the syllabus, most exam questions will use equals for simplicity.

Worked Example

Prove that open parentheses 2 x minus 2 close parentheses open parentheses x minus 3 close parentheses plus 2 left parenthesis x minus 1 right parenthesis equals 2 left parenthesis x minus 2 right parenthesis left parenthesis x minus 1 right parenthesis for all x element of straight real numbers.

1-4-1-aa-sl-language-of-proof-we-solution

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Proof by Deduction

What is proof by deduction?

  • Proof by deduction means proving that a statement using common mathematical results

    • For example

      • using algebra

      • using geometry

How do I apply proof by deduction to integers?

  • To prove results about integers (whole numbers), you need to first represent the integers as algebraic letters or terms

    • The following table shows the most commonly used algebraic terms

Type of integer

Term

Comment

Any integer

n

Consecutive integers

n comma space space n plus 1

This means one after the other. Could also use n minus 1 comma space space n

Any two integers

n comma space space m

A different letter is used (to show it is not necessarily consecutive)

An even integer

2 n

Consecutive even integers

2 n comma space space 2 n plus 2

Could also use 2 n minus 2 comma space space 2 n

Any two even integers

2 n comma space space 2 m

An odd integer

2 n plus 1

Could also use 2 n minus 1

A multiple of 5

5 n

A multiple of k

k n

One more than a multiple of 3

3 n plus 1

A square number

n squared

A cube number

n cubed

A rational number

a over b

Where a and b are integers and b not equal to 0

  • You then need to be able to apply operations to the terms above

    • Common operations are the

      • sum (plus)

      • difference (minus)

      • product (cross times)

      • square open parentheses... close parentheses squared

How do I show that a result is odd or even?

  • To prove an expression is even, show that it can be written as Error converting from MathML to accessible text.

    • For example, 2 open parentheses n squared minus 3 n close parentheses is even

      • This may require factorising out a 2

  • To prove something is odd, show that it can be written as Error converting from MathML to accessible text.

    • For example, 2 open parentheses n plus m close parentheses plus 1 is odd

  • Make sure the part inside the brackets is an integer

    • For example, 2 open parentheses n plus 1 third close parentheses is not even as 1 third is not an integer

  • You can apply similar ideas to prove expressions are multiples of other numbers

    • For example, 7 open parentheses n squared plus 2 n close parentheses is a multiple of 7

Worked Example

Prove that the sum of any two consecutive odd numbers is always even.

1-4-1-aa-sl-proof-by-deduction-we-solution

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    Author
    Reviewer
    Amber

    Author: Amber

    Expertise: Maths Content Creator

    Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.

    Mark Curtis

    Reviewer: Mark Curtis

    Expertise: Maths Content Creator

    Mark graduated twice from the University of Oxford: once in 2009 with a First in Mathematics, then again in 2013 with a PhD (DPhil) in Mathematics. He has had nine successful years as a secondary school teacher, specialising in A-Level Further Maths and running extension classes for Oxbridge Maths applicants. Alongside his teaching, he has written five internal textbooks, introduced new spiralling school curriculums and trained other Maths teachers through outreach programmes.