Writing down solutions notation

Question

How do you write solutions to a simple equation in set notation?

For example: - the solution to $x-4=0$ - the solutions to $x^2-x=0$?

Thanks in advance.

Answer 1

If the solution to your equation is $a_{1},...a_{m}$ then in set notation you simply write $$ \{a_{1},...,a_{m}\} $$

in set notation

Answer 2

Well, you can list the elements of the set of solutions: For your two examples, you can use $\lbrace 4\rbrace$ and $\lbrace 0,1\rbrace$.

Answer 3

You can write down the solutions both explicitly or implicitly, the latter uses the notation $\{x|\ldots\}$ or $\{x:\ldots\}$ which both means “the set of numbers $x$ such that $\ldots$ is true”. Here are some examples:

$$\begin{array}{c|c|c} \text{Equations} &\text{(Implicit) Set of solutions} & \text{(Explicit) Set of solutions} \\ \hline \overset{\displaystyle \color{white}d\,}{}3x+2=3x-1&\overset{\displaystyle \color{white}d\,}{}\{x:3x+2=3x-1\}& \varnothing \\ \sin x=-\sin(-x) &\{x:\sin x=-\sin(-x)\}&\mathbb R\\ x^2-3x=-2 &\{x\mid x^2-3x=-2\}& \{1,2\} \\ 6x+15y=39 &\{(x,y)\mid 6x+15y=39\}&\left\{\left(x,\frac{13-2x}5\right)\,\left|\,\right.x\in\mathbb R\right\} \\ \end{array}$$