Extension Proof – The nature of

This lesson comprises two (2) master classes focusing on:

• Language of proof
• Proof by contradiction
• Examples and counter-examples
• Inequality proofs

Content:

MEX-P1

• Use the formal language of proof, including the terms statement, implication, converse, negation and contrapositive
  • use the symbols for implication (\( \implies \)), equivalence (\( \leftrightarrow \)) and equality (\( = \)) , demonstrating a clear understanding of the difference between them
  • use the phrases ‘for all’ (\( \forall \)), ‘if and only if’ (\( iff \)) and ‘there exists’ (\( \exists \))
  • understand that a statement is equivalent to its contrapositive but that the converse of a true statement may not be true
• Prove simple results involving numbers
• Use proof by contradiction including proving the irrationality for numbers such as \( \sqrt{2} \) and \( \log_2{5} \)
• Use examples and counter-examples
• Prove results involving inequalities. For example:
  • prove inequalities by using the definition of \( a>b \) for real \( a \) and \( b \)
  • prove inequalities by using the property that squares of real numbers are non-negative
  • prove and use the triangle inequality \( |x|+|y| \ge |x+y| \) and interpret the inequality geometrically
  • establish and use the relationship between the arithmetic mean and geometric mean for two non-negative numbers
• Prove further results involving inequalities by logical use of previously obtained inequalities