Language of Proof Revision Notes for Edexcel A-Level Mathematics

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:

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.

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 $x^2 = 4 .$ "

Proof Techniques:

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

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Example: Prove that the sum of two odd numbers is even.

Let $a = 2m + 1$ and $b = 2n + 1$ , where $m$ and $n$ are integers.
Then, $a + b = (2m + 1) + (2n + 1) = 2(m + n + 1)$ , which is even.

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

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Example: Prove that a number less than $4$ is either $1, 2,$ or $3$ .

Consider all integers less than $4$ : $1, 2, 3$ . In each case, the statement holds.

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

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Example: Disprove the statement "All prime numbers are odd."

Counterexample: $2$ is a prime number and it is even, so the statement is false.

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.

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Example: Prove that $\sqrt{2}$ is irrational.

Assume $\sqrt{2}$ is rational, so it can be expressed as $\ \frac{p}{q}$ in its lowest terms.
Then $2 = \frac{p^2}{q^2}$ , implying $p^2 = 2q^2$ , so $p$ must be even.
Let $p = 2k$ , then $4k^2 = 2q^2$ implies $\ q^2 = 2k^2 ,$ so $q$ must also be even.
This contradicts the assumption that $\frac{p}{q}$ is in its lowest terms, so $\ \sqrt{2}$ is irrational.

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 $\geq 1$ )

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Example: $\mathbb{N} = \{ 1, 2, 3, 4, \ldots \}$

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

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Example: $\mathbb{Z} = \{ \ldots, -3, -2, -1, 0, 1, 2, 3, \ldots \}$

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

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Definition: $\mathbb{Q} = \left\{ \frac{a}{b} : a, b \in \mathbb{Z} \right\}$ $a$ and $b$ are integers

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

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Definition: $\mathbb{C} = \{ a + bi : a, b \in \mathbb{R} \}$

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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Examples:

List all numbers that are members of the following sets:

a) $\{ x \mid x \leq 4, x \in \mathbb{N} \}$
$( x )$ is less than or equal to $4$ , ( $x$ ) is a natural number.
Answer: $\{ 1, 2, 3, 4 \}$
b) $\{ x : x \leq 4, x \in \mathbb{N} \} \cap \{ p : p \text{ is prime} \}$
Find where the two sets overlap.
$\{ 1, 2, 3, 4 \} \cap \{ 2, 3, 5, 7, 11, 13, \ldots \}$
Answer: $\{ 2, 3 \}$
c) $\{ x : x \leq 4, x \in \mathbb{N} \} \cap \{ 2n : n \in \mathbb{N} \}$
Even numbers in $\mathbb{N}$ less than or equal to $4$ .
$\{ 1, 2, 3, 4 \} \cap \{ 2, 4, 6, 8, \ldots \}$
Answer: $\{ 2, 4 \}$

Interval Notation

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

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Example 1:

$[6, 12]$ : All real numbers between $6$ and $12$ inclusive.
Equivalent set notation: $\{ x : x \in \mathbb{R}, 6 \leq x \leq 12 \}$ Example 2:
$(7, 14]$ : All real numbers between $7$ and $14$ , $7$ not included, $14$ included.
Equivalent set notation: $\{ x : 7 < x \leq 14, x \in \mathbb{R} \}$

Incorrect Intervals:

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Example 3:

$(12, 8)$
Issue: Numbers in the wrong order.

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Example 4:

$(7, \infty)$
Issue: Nothing can equal infinity.

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 $f(x) = x^2 - x + 2$ $f (x) = x^{2} - x + 2$ for each value of $x$ $x$ in the set $\{0, 1, 2\}$ ${0, 1, 2}$ .
- For $x = 0$ , $f(0) = 0^2 - 0 + 2 =$ 2.
- For $x = 1$ , $f(1) = 1^2 - 1 + 2 =$ 2.
- For $x = 2$ , $f(2) = 2^2 - 2 + 2 =$ 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.

Language of Proof (Edexcel A-Level Mathematics): Revision Notes

1.1.1 Language of Proof