NSW Syllabuses

# Mathematics Extension 2 Stage 6 - Year 12 Extension 2 - Proof MEX-P1 The Nature of Proof

## Outcomes

#### A student:

• MEX12-1

understands and uses different representations of numbers and functions to model, prove results and find solutions to problems in a variety of contexts

• MEX12-2

chooses appropriate strategies to construct arguments and proofs in both practical and abstract settings

• MEX12-7

applies various mathematical techniques and concepts to model and solve structured, unstructured and multi-step problems

• MEX12-8

communicates and justifies abstract ideas and relationships using appropriate language, notation and logical argument

## Subtopic Focus

The principal focus of this subtopic is to develop rigorous mathematical arguments and proofs specifically in the context of number.

Students develop an appreciation of the necessity for rigorous and robust methods to prove the validity of a variety of concepts related to number.

## Content

• Students:
• use the symbols for implication $$\left(\Rightarrow\right)$$, equivalence $$\left(\Leftrightarrow\right)$$ and equality $$\left(=\right)$$, demonstrating a clear understanding of the difference between them (ACMSM026)
• use the phrases ‘for all’, 'if and only if' and ‘there exists’ (ACMSM027)
• understand that a statement is equivalent to its contrapositive but that the converse of a true statement may not be true
• prove irrationality by contradiction for numbers including but not limited to $$\sqrt2$$ and $$\log_25$$ (ACMSM063)
• use examples and counter-examples (ACMSM028)
• prove simple results involving numbers, including but not limited to the product of three consecutive integers being divisible by 6  (ACMSM061)
• prove results involving inequalities
• prove linear or quadratic inequalities by using the definition of $$a>b$$ for real $$a$$ and $$b$$
• prove linear or quadratic inequalities by using the property that squares of real numbers are non-negative
• prove and use the triangle inequality $$\left|x\right|+\left|y\right| \ge\left|x+y\right|$$ and interpret the inequality geometrically
• establish and use the relationship between the arithmetic mean and geometric mean for two non-negative numbers