Nature of Proof Key Terms: A Comprehensive Glossary

Introduction

The Nature of Proof module is unique because it tests your ability to form logical arguments just as much as your algebraic skills. In HSC Mathematics Extension 2, you are introduced to the formal language of mathematical logic. This glossary provides the definitive mathematical definitions and illustrative examples for the core terminology used throughout the Proof syllabus.

Executive Summary

This guide serves as a quick-reference dictionary for:

Logical Statements: Propositions, implications, negations, converses, and contrapositives.
Numbers and Sequences: Integers, rational numbers, arithmetic mean, geometric mean, recurrence relations.
Methods of Proof: Mathematical induction, proof by contradiction, counterexamples.
Inequalities: The squeeze theorem, triangle inequality.

What is this about?

When you construct a mathematical proof, using the correct terminology is mandatory for full marks. Knowing the precise difference between the "converse" and the "contrapositive" of a statement dictates whether your proof is logically valid. This page ensures you understand exactly what the examiner means when they ask you to "negate the following proposition using quantifiers."

Main Content: Key Terms

A–E

Arithmetic Mean: The mathematical average of a set of numbers, calculated by adding them together and dividing by the count.
- Example: The arithmetic mean of $4$ and $16$ is $\frac{4+16}{2} = 10$ .
Binomial Theorem: An algebraic formula for expanding powers of binomials, stating $(x+y)^n = \sum_{k=0}^n \binom{n}{k} x^{n-k} y^k$ $(x + y)^{n} = \sum_{k = 0}^{n} (k n) x^{n - k} y^{k}$ .
- Example: $(x+y)^3 = x^3 + 3x^2y + 3xy^2 + y^3$ .
Contrapositive: A logical statement formed by negating and swapping both the hypothesis and conclusion of an implication. If the original is $P \implies Q$ $P ⟹ Q$ , the contrapositive is $\sim Q \implies \sim P$ $\sim Q ⟹ \sim P$ . It is always logically equivalent to the original statement.
- Example: Original: "If it rains, the grass is wet." Contrapositive: "If the grass is not wet, then it did not rain."
Converse: A statement formed by swapping the hypothesis and conclusion. The converse of $P \implies Q$ $P ⟹ Q$ is $Q \implies P$ $Q ⟹ P$ . It is not necessarily logically equivalent to the original.
- Example: Original: "If it is a square, it has four sides." Converse: "If it has four sides, it is a square." (False, it could be a rectangle).
Counterexample: A specific case that demonstrates a universal statement is false.
- Example: To disprove "All prime numbers are odd", the counterexample is the number $2$ .
Equivalent: Two statements are logically equivalent if they share the exact same truth value in every scenario, denoted by $P \iff Q$ $P ⟺ Q$ .
- Example: $x^2 = 9$ is equivalent to $x = 3 \text{ or } x = -3$ .

G–N

Geometric Mean: The $n$ $n$ th root of the product of $n$ $n$ numbers. For two numbers, it is the square root of their product.
- Example: The geometric mean of $4$ and $16$ is $\sqrt{4 \times 16} = \sqrt{64} = 8$ .
Implication: An "if-then" logical relationship where the truth of a hypothesis guarantees the truth of a conclusion, denoted by $P \implies Q$ $P ⟹ Q$ .
- Example: If $x = 3$ , then $x^2 = 9$ .
Inequality: A mathematical statement comparing the relative size of two expressions using $<, >, \le,$ $<, >, \leq,$ or $\ge$ $\geq$ .
- Example: $x^2 + y^2 \ge 2xy$ for all real numbers.
Integer: A whole number that can be positive, negative, or zero, belonging to the set $\mathbb{Z}$ $Z$ .
- Example: $-5, 0, \text{ and } 42$ are integers. $2.5$ is not.
Mathematical Induction: A method of proof that establishes the truth of an infinite sequence of statements. It involves proving a base case (e.g., $n=1$ $n = 1$ ) and an inductive step (if true for $k$ $k$ , then true for $k+1$ $k + 1$ ).
- Example: Proving that the sum of the first $n$ odd numbers is $n^2$ for all integers $n \ge 1$ .
Negation: The logical opposite of a statement, denoted by $\sim P$ $\sim P$ . If $P$ $P$ is true, $\sim P$ $\sim P$ is false.
- Example: If $P$ is "All birds can fly", the negation $\sim P$ is "There exists at least one bird that cannot fly."

P–S

Proof: A rigorous, deductive logical argument demonstrating that a mathematical proposition is invariably true based on accepted axioms.
- Example: Expanding $(a-b)^2 \ge 0$ to formally prove $a^2 + b^2 \ge 2ab$ .
Proof by Contradiction: A method where you assume the negation of what you want to prove is true, and logically deduce a mathematical impossibility (a contradiction), meaning the original statement must be true.
- Example: Assuming $\sqrt{2} = \frac{a}{b}$ (a rational fraction in simplest terms) and proving that this forces both $a$ and $b$ to be even, contradicting the "simplest terms" assumption.
Proposition: A declarative sentence that is either objectively true or objectively false, but not both.
- Example: "The Earth has two moons" is a false proposition. " $x + 5 = 10$ " is not a proposition until the value of $x$ is defined.
Quantifier: Symbols that specify the scope of a variable in a logical statement. "For all" ( $\forall$ $\forall$ ) is the universal quantifier. "There exists" ( $\exists$ $\exists$ ) is the existential quantifier.
- Example: $\forall x \in \mathbb{R}, x^2 \ge 0$ (For all real numbers $x$ , $x^2$ is greater than or equal to zero).
Rational Number: Any number that can be expressed as a fraction $\frac{p}{q}$ $\frac{p}{q}$ , where $p$ $p$ and $q$ $q$ are integers and $q \neq 0$ $q \neq = 0$ . Belongs to the set $\mathbb{Q}$ $Q$ .
- Example: $0.75$ is a rational number because it can be written as $\frac{3}{4}$ . $\pi$ is not.
Recurrence Relation / Recursive Formula: An equation that defines the next term in a sequence based on one or more previous terms.
- Example: $T_{n} = T_{n-1} + T_{n-2}$ (The Fibonacci sequence).
Squeeze Theorem: A theorem used to evaluate the limit of a function by "trapping" it between two other functions that both converge to the same limit.
- Example: Because $-x^2 \le x^2 \sin(\frac{1}{x}) \le x^2$ , and both $-x^2$ and $x^2$ approach $0$ as $x \to 0$ , the limit of $x^2 \sin(\frac{1}{x})$ must also be $0$ .
Statement: Interchangeable with "Proposition" in this syllabus. A mathematical sentence that is definitively true or false.
- Example: "7 is a prime number."

T

Triangle Inequality: A theorem stating that the absolute value of a sum is always less than or equal to the sum of the absolute values: $|x + y| \le |x| + |y|$ $∣ x + y ∣ \leq ∣ x ∣ + ∣ y ∣$ . Geometrically, the length of any side of a triangle is less than the sum of the other two sides.
- Example: $|5 + (-3)| \le |5| + |-3| \implies 2 \le 8$ .

mini-FAQ page

Q: Will I be asked to define these terms in an exam? A: You won't be asked for dictionary definitions, but you will be asked to "Write the contrapositive of the statement" or "Use a counterexample to disprove...". Knowing the definition is required to execute the math.

Q: Is there a difference between a proposition and a statement? A: In the context of the HSC syllabus, they are used interchangeably to mean a sentence with a definitive truth value.

Common mistakes to avoid

Confusing Converse and Contrapositive: If asked to prove a statement by its contrapositive, do not write the converse. The converse is often false even when the original statement is true!
Negating universal statements incorrectly: The negation of "All apples are red" is NOT "No apples are red." The correct negation is "At least one apple is not red" (There exists an apple that is not red).

Practice on Vu's Maths Hub

Language precision is the key to conquering the Nature of Proof.

Practice translating English into logical quantifiers in the HSC Proof Booklet.
Test your understanding of converses and contradictions at Vu's Maths Hub.

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Want to master Proofs and guarantee top marks in HSC Mathematics Extension 2? Visit Vu's Maths Hub for in-depth booklets, rigorous worked solutions, and expert advice to help you ace your exams!