Proof by deduction

In a nutshell

Proof by deduction (or direct proof) involves proving a statement to be true using algebra. There are different expressions that you need to know which help represent numbers in maths.

Expressions for proof

Here are some expressions for numbers that you need to know.

DESCRIPTION	EXPRESSION
An even number	$2n$
An odd number	$2n+1$
A multiple of $x$	$xn$
A square number	$n^2$
Two consecutive numbers	$n,n+1$
Two random numbers	$n,m$

Example 1

Represent two consecutive odd squares algebraically.

Work step by step, going through the description.

Represent two consecutive numbers:

$n,n+1$

These numbers have to be odd, so double them (making them even), then add $1$ :

$2(n)+1,2(n+1)+1$

$2n+1,2n+3$

These numbers have to be square numbers, so square them:

$(2n+1)^2,(2n+3)^2$

Two consecutive odd squares are $\underline{ (2n+1)^2}$ and $\underline {(2n+3)^2}$ .

Algebraic proof

To prove something algebraically, use the expressions to represent each number with algebra. Then, manipulate the expression until it's in the desired form.

Example 2

Prove that the difference between two consecutive odd squares is always a multiple of $8$ .

Two consecutive odd squares can be represented as:

$(2n+1)^2,(2n+3)^2$

As shown in the above example. Their difference can be found by expanding the brackets and simplifying.

$\begin{aligned}(2n+3)^2-(2n+1)^2&=(2n+3)(2n+3)-(2n+1)(2n+1)\\&=(4n^2+12n+9)-(4n^2+4n+1)\\&=8n+8\\&=8(n+1)\end{aligned}$

The expression simplifies to $8(n+1)$ , which is a multiple of $8$ because it is in the form $8\times$ an integer.

Therefore, the difference between two consecutive squares is always a multiple of eight.

Proof with inequalities

Algebraically manipulating inequalities is the same as algebraically manipulating equations. The important rule to remember is that when multiplying or dividing by a negative number, reverse the inequality symbol.

To prove a statement that involves an inequality, it's important to remember the order of logic.

Note: You may have to use the $\Rightarrow$  symbol to formally show your reasoning.

Example 3

Prove that $2x+1$ is always positive for all positive values of $x$ .

Rewrite the statement using "if _, then _":

"If $x$ is positive, then $2x+1$ is positive"

Rewrite it using arrow notation:

$x>0\Rightarrow 2x+1>0$

This means that to prove this formally, begin with the left hand side and use algebra to end up at the right hand side:

$\begin{aligned}x>0&\Rightarrow 2x>0\\&\Rightarrow 2x+1>1>0\\&\Rightarrow 2x+1>0\end{aligned}$

$\underline{x>0\Rightarrow 2x+1>0}$

Inequality proofs with square numbers

Sometimes, you may be asked to prove a result that will require you to use a certain property of square numbers:

Square numbers are positive

Example 4

Prove that, for any value of $x$ , $x^2\geq 2x-1$ .

First, do some rough work to see how to begin the proof.

Try to experiment with the expression:

$x^2\ge 2x-1\Rightarrow x^2-2x+1\ge 0\Rightarrow (x-1)(x-1)\ge0\Rightarrow (x-1)^2\ge 0$ ( $\star$ )

This means that the proof uses the property of square numbers being either positive or $0$ :

$x^2 \ge 0$

Begin the formal proof using the steps used in the rough work, but in the "correct" direction:

For any value of $x$ , $(x-1)^2\ge 0$ because it is a square.

$\begin{aligned}(x-1)^2\ge 0&\Rightarrow (x-1)(x-1)\ge 0\\&\Rightarrow x^2-2x+1 \ge 0\\&\Rightarrow x^2 \ge 2x-1\end{aligned}$

Therefore, for any value of $x$ , $\underline{x^2 \ge 2x-1}$ .

Note: The reason why the working out at ( $\star$ ) is not correct is that you should always aim to end your proof with the expression that you are trying to prove.