Home · Unit 1 · Topic 2

Topic 2: Introduction to proof

The nature of proof

5 hours

• Use implication, converse, equivalence, negation, contrapositive.
• Use proof by contradiction.
• Use the symbols for implication (⇒), equivalence (⟺), and equality (=).
• Use the quantifiers ‘for all’ (∀) and ‘there exists’ (∃).
• Define and use set notation of number systems, including integers (ℤ), positive integers (ℤ⁺), negative integers (ℤ⁻), rational numbers (ℚ), irrational numbers (ℚ′), and real numbers (ℝ).
• Use the set notation symbol ‘is an element of’ (∈).
• Use examples and counterexamples.

Rational and irrational numbers

5 hours

• Prove results involving integers, e.g. proving that the product of two consecutive odd numbers is an odd number and 5n² + 3n + 6 ∀ n ∈ ℤ is an even number.
• Express rational numbers as terminating or eventually recurring decimals and vice versa.
• Prove irrationality by contradiction. 1 interactive