Proof

Skill 1: Proof Notation

The following notation will be used throughout proofs, but will also be relevant to topics later in the course.

A set is a collection of objects or numbers (called elements). A set is denoted using a capital letter, and curly brackets are used to show what is in the set, e.g. $A = {3,4,5 }$.

You can write sets in various different ways:

• as a list of elements: e.g. ${3,4,5}$
• as a rule: e.g. ${text{prime numbers}$$text{ between 0 and 10}}$
• as mathematical notation: e.g. ${x : x > 5 }$ – this means “the set of values such that $x$ is greater than $5$”

There are different variations of an $=$ sign that you need to be aware of:

• $neq$ means “not equal to”
• $approx$ means “approximately equal to”
• $equiv$ means two things are “equivalent”. It is called the identity symbol.

There are logic symbols that you need to understand:

• “$p Rightarrow q$” means “if $p$ then $q$” or “$p$ implies $q$”
• “$p iff q$” means “$p$ if and only if $q$” or “$p$ implies $q$ and $q$ implies $p$”

Note: “if and only if” can be written as “iff”

Skill 2: Direct Proof

A direct proof (or proof by deduction) is a proof where a statement is proven to be true using fundamental mathematical principles.

Example: Prove that $n^2 – 6n + 11$ is positive for any integer.

Completing the square gives

$textcolor{red}{(n-3)^2} + textcolor{blue}{2}$

$textcolor{red}{(n-3)^2}$ is always positive, since it is a square number.

Therefore, if we add $textcolor{blue}{2}$ to the square number, then the result is still positive.

Hence, $n^2 – 6n + 11$ is positive for any integer.

Skill 3: Proof by Exhaustion

In a proof by exhaustion, you will need to break situations down into two or more cases and try all possibilities to prove that the statement holds true for each case.

Example: Prove the following statement:

“For any integer $x$, $f(x)=3x^2+3x-2$ is an even integer”

We need to split the statement into two cases: if $x$ is an even number, or if $x$ is an odd number.

Case 1: If $x$ is even, then $textcolor{red}{x=2n}$ for some integer $n$.

Then,

$begin{aligned} f(2n) &= 3(2n)^2+3(2n)-2 &= 12n^2 + 6n – 2 &= 2(6n^2+3n-1) end{aligned}$

$n$ is an integer, therefore $6n^2+3n-1$ is an integer.

Hence, $2(6n^2+3n-1)$ is an even integer.

So, $f(x)$ is even when $x$ is even.

Case 2: If $x$ is an odd number, then $textcolor{blue}{x = 2m+1}$ for some integer $m$.

Then,

$begin{aligned} f(2m+1) &= 3(2m+1)^2 + 3(2m+1) – 2 &= 12m^2 + 12m + 3 + 6m + 3 – 2 &= 12m^2 + 18m + 4 &= 2(6m^2 + 9m + 2) end{aligned}$

$m$ is an integer, therefore $6m^2 + 9m + 2$ is an integer.

Hence, $2(6m^2 + 9m + 2)$ is an even integer.

So, $f(x)$ is even when $x$ is odd.

Hence, $f(x)$ is even for any integer $x$, and the statement is true.

Skill 4: Disproof by Counter-Example

To disprove a statement by a counter-example, all you need to do is show that the statement is false for one case.

Example: Disprove the following statement:

“For any pair of real numbers $x$ and $y$, if $x^2=y^2$ then $x=y$”

We only need to find one set of values such that the statement is false.

For example, take $x=3$ and $y = -3$

Then,

$x^2 = 3^2 = 9$ and $y^2 = (-3)^2 = 9$

So, $textcolor{limegreen}{x^2 = y^2}$

However, $textcolor{red}{x neq y}$, and so the statement is not true.