1.1.1 Language of Proof

Language of Proof

In mathematics, proof is the process of establishing the truth of a statement through logical reasoning. The language of proof is fundamental for you to know for A-Level mathematics, as it is used to validate mathematical statements rigorously.

Key Components of the Language of Proof:

1. Statements:

• Theorems: Proven statements that are based on established truths like axioms or other theorems.
• Lemmas: Supporting theorems that assist in proving more significant results.
• Corollaries: Consequences that directly follow from a theorem.
• Conjectures: Statements believed to be true but not yet proven.

2. Quantifiers:

• Universal quantifier $\forall$: Indicates that a statement applies to all elements in a set. For example, "For all $x$ in the set of real numbers, $x^2 \geq 0 .$"
• Existential quantifier $\exists :$ Indicates that there is at least one element in the set for which the statement is true. For example, "There exists an integer $\ x$ such that $x^2 = 4 .$"

3. Proof Techniques:

• Proof by Deduction: Definition: Start from general principles or known facts and logically deduce the statement you want to prove.

• Proof by Exhaustion: Definition: Prove a statement by considering all possible cases and showing that the statement holds in each case.

• Proof by Counterexample: Definition: Disprove a statement by providing a specific example where the statement does not hold.

• Proof by Contradiction: Definition: Assume the opposite of what you want to prove, and show that this assumption leads to a contradiction, thereby proving the original statement.

Set Notation

• Introduction:
  • When dealing with proofs, it is often the case that we need to work within specific sets of numbers.
• Common Sets:
  • $\mathbb{N}$ : Set of natural numbers (all positive integers $\geq 1$)

• $\mathbb{R}$ : Set of real numbers
• $\mathbb{Z}$ : Set of all integers (positive, negative, and zero)

• $\mathbb{Q}$ : Set of all rational numbers

• $\mathbb{C}$ : Set of complex numbers

Definition of a Set:

• $A$ set is a "bag" containing "things".
• Curved points (braces) indicate a set: $\{\}$
• No repeated items in sets. For example, {1, 2, 3} is correct, not {1, 2, 2, 3}.

Notation:

• $\{\}$ : A set.
• $\in$ : "is a member of".
• $\emptyset$ or $\{\}$ : An empty set.
• $\cup$ : Union of two sets, combining elements and removing repetitions.
• $\cap$ : Intersection of two sets, elements they have in common.
• $\complement$ : The complement of a set, all items that are not in the set.

Common Sets:

• $\mathbb{N}$: Natural numbers $\{ 1, 2, 3, \ldots \}$.
• $\mathbb{Z}$ : Integers $\{ \ldots, -3, -2, -1, 0, 1, 2, 3, \ldots \}$.
• $\mathbb{N}_0$ : Natural numbers including $0$.
• $\mathbb{R}$ : Real numbers.
• $\mathbb{Q}$ : Rational numbers $\left\{ \frac{a}{b} : a, b \in \mathbb{Z}, \right\}$. Note: $\frac{a}{b}$ means "such that" a and b are integers.

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Interval Notation

• Purpose: Convenient way of expressing inequalities.
• Symbols:
  • Square brackets: Means "can be equal to."
  • Round brackets: Means "cannot equal."

Incorrect Intervals:

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Past Exam Question:

June 2018, Paper 1, Question 8: "Prove by exhaustion that the function $f(x) = x^2 - x + 2$ is always positive for integer values of $x$ in the set $\{0, 1, 2\}$."

Solution Outline:

• Statement: Evaluate $f(x) = x^2 - x + 2$ for each value of $x$ in the set $\{0, 1, 2\} .$
  • For $( x = 0 ), \ f(0) = 0^2 - 0 + 2 = :success[2] .$
  • For $( x = 1 ), \ f(1) = 1^2 - 1 + 2 = :success[2] .$
  • For $( x = 2 ), \ f(2) = 2^2 - 2 + 2 = :success[4]$.
• Conclusion: Since $f(x)$ is positive for all values of $\ x$ in the set $\ \{0, 1, 2\} ,$ the function is always positive. This example demonstrates proof by exhaustion, where all possible cases are checked individually to establish the truth of the statement.

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