Mathematical structure and arguments

In a nutshell

There are different types of mathematical structures that you need to know: the main two being equations and identities. When working with mathematical arguments - such as manipulating equations algebraically - it is important to be able to identify and work with the logic behind the steps.

Equations and identities

Definitions

EQUATION	An algebraic identity that is only true for certain values of an unknown variable.
IDENTITY	An algebraic identity that is always true for every value of the unknown variable. Identities are represented with the $\equiv$ symbol.

Note: The unknown variable is usually $x$ .

Example 1

$2x=4$ is an equation because it is only true for one value ( $x=2$ ).

$x^2+3x\equiv x(x+3)$ is an identity because it is true for every value of $x$ .

Prepositional logic

Prepositional logic is often used in formal mathematics to prove something. There are three main symbols that you need to know.

SYMBOL	MEANING	EXAMPLE
$A\Rightarrow B$	" $A$ implies $B$ "/"If $A$ , then $B$ "	The shape is a square $\Rightarrow$ the shape has four sides ("If the shape is a square, then it must have four sides")
$A\Leftarrow B$	" $A$ is implied by $B$ "/"If $B$ , then $A$ "	The angle is reflex $\Leftarrow$ the angle is between $180\degree$ and $360\degree$ ("If the angle is between $180\degree$ and $360\degree$ , then the angle is reflex")
$A \iff B$	$A\Rightarrow B$ and $B\Rightarrow A$	$x$ is positive $\iff$ $2x$ is positive ("If $x$ is positive, then $2x$ is positive AND if $2x$ is positive, then $x$ is positive")

Note: $A \iff B$ is sometimes referred to as " $A$ if and only if $B$ " or " $A$ iff $B$ ".

The notation is very important. It would be incorrect to say:

The shape is a square $\Leftarrow$ the shape has four sides

Just because a shape has four sides, does not mean that the shape is a square. The shape could be a rectangle, parallelogram, etc.

Example 2

Prove that $x=3\Rightarrow x^2=9$ . Is the reverse direction also true?

Start from the left hand side, and use the correct arrow and logic to arrive at the right hand side:

$\begin{aligned}x=3&\Rightarrow x^2=3^2\\&\Rightarrow x^2=9\end{aligned}$ 

Hence, the statement has been proved.

The reverse direction refers to the statement:

$x=3\Leftarrow x^2=9$ 

This time, start from the right hand side, and use the correct logic and algebra to arrive at the left hand side:

$\begin{aligned}x^2=9&\Rightarrow x^2-9=0\\&\Rightarrow (x+3)(x-3)=0\\&\Rightarrow x=\pm3 \end{aligned}$ 

The reverse direction is not true because of the extra solution: $x=-3$ .