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

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.

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: $$x=-3$$.