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:
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 are often infinite sets; we can't list all the numbers. So, we use a special notation.
Example:
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.