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.