Proof

use the formal language of proof, including the terms statement, implication, converse, negation and contrapositive (ACMSM024)ACMSM024

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• use the symbols for implication $(\Rightarrow )$, equivalence $(\iff )$ and equality $(=)$ , demonstrating a clear understanding of the difference between them (ACMSM026)ACMSM026

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• use the phrases ‘for all’ $(\forall )$, ‘if and only if’ $(\mathrm{iff})$ and ‘there exists’ $(\exists )$ (ACMSM027)ACMSM027

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• understand that a statement is equivalent to its contrapositive but that the converse of a true statement may not be true

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prove simple results involving numbers (ACMSM061)ACMSM061

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use proof by contradiction including proving the irrationality for numbers such as $\sqrt{2}$ and ${\log }_{2}5$ (ACMSM025, ACMSM063)ACMSM025 ACMSM063

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use examples and counter-examples (ACMSM028)ACMSM028

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prove results involving inequalities. For example:

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• prove inequalities by using the definition of $a>b$ for real $a$ and $b$

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• prove inequalities by using the property that squares of real numbers are non-negative

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• prove and use the triangle inequality $\left|x\right|+\left|y\right|\ge \left|x+y\right|$ and interpret the inequality geometrically

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• establish and use the relationship between the arithmetic mean and geometric mean for two non-negative numbers

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prove further results involving inequalities by logical use of previously obtained inequalities

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prove results using mathematical induction where the initial value of $n$ is greater than 1, and/or $n$ does not increase strictly by 1, for example prove that $n^2+2n$ is a multiple of 8 if $n$ is an even positive integer

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understand and use sigma notation to prove results for sums, for example: ${\sum }_{n=1}^N\frac{1}{\left(2n+1\right)\left(2n-1\right)}=\frac{N}{2N+1}$

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understand and prove results using mathematical induction, including inequalities and results in algebra, calculus, probability and geometry. For example:

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• prove inequality results, eg $2^n>n^2$, for positive integers $n>4$

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• prove divisibility results, eg $3^{2n+4}-2^{2n}$ is divisible by 5 for any positive integer $n$

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• prove results in calculus, eg prove that for any positive integer $n$ , $\frac{d}{dx}(x^n)=n{x}^{n-1}$

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• prove results related to probability, eg the binomial theorem: ${\left(x+a\right)}^n={\sum }_{r=0}^n{}^n{C}_r{x}^{n-r}{a}^r$

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• prove geometric results, eg prove that the sum of the exterior angles of an $n$-sided plane convex polygon is 360°

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use mathematical induction to prove first-order recursive formulae

not written yet · needs a worked example