Solution Sets
Solution Sets for Equations
The set containing all the solutions of an equation is called the solution set for that equation.
If an equation has no solutions, we write $\varnothing $ for the solution set. $\varnothing $ means the null set (or empty set).
Equation

Solution Set

$3x+5=11$

$\left\{2\right\}$

${x}^{2}=x$

$\{0,1\}$

$x+1=1+x$

$\text{R}$
(the set of all real numbers)

$x+1=x$

$\varnothing $
(the empty set)

Sometimes, you may be given a replacement set, and asked to test whether the equation is true for all values in the replacement set.
Example 1:
Find the solution set for the equation $z+z=z\times z$ if the replacement set is $\{0,1,2,3,\}$ .
One method of solving this problem is to test all the values in the replacement set using a table.
$\begin{array}{ccc}\hline z& z+z=z\times z& \text{Result}\\ 0& \begin{array}{l}0+0\stackrel{?}{=}0\times 0\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0=0\end{array}& \\ 1& \begin{array}{l}1+1\stackrel{?}{=}1\times 1\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}2\ne 1\end{array}& \\ 2& \begin{array}{l}2+2\stackrel{?}{=}2\times 2\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}4=4\end{array}& \\ 3& \begin{array}{l}3+3\stackrel{?}{=}3\times 3\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}6\ne 9\end{array}& \\ \hline\end{array}$So, the solution set for this equation is $\{0,2\}$ .
Solution Sets for Inequalities
Solution sets for inequalities are often infinite sets; we can't list all the numbers. So, we use a special notation.
Example 2:
Solve the inequality
$x+2>3$ .
By subtracting 2 from both sides, we get the equivalent inequality
$x>5$ .
So, the solution set is
$\left\{x\text{\hspace{0.17em}}\right\text{\hspace{0.17em}}x>5\}$ .
(To read this, you would say: " $x$ such that $x$ is greater than negative five." The  symbol means "such that" in this case.)
Often, the solutions to inequalities are also written in interval notation .