Set notation and interval notation

In a nutshell

When solving a linear inequality, the answer obtained is usually also written as an inequality. These inequalities can be written using set notation or interval notation. 

Set notation and interval notation

When an answer is an inequality given in the form $$x \geq 7$$, it can be written in the set notation $$\{x : x \geq 7\}$$. This is read as '$$x$$ such that $$x$$ is greater than or equal to $$7$$'.

This can also be written as an interval notation: $$[7,\ \infin)$$​. This means that the solution lies in the interval from $$7$$ to $$\infin$$. Unless otherwise stated, $$x$$ is assumed to be a real number, therefore $$\infin$$ is not included in the interval.​

Common notation

Example 1

Write the following in set notation:

i) $$x \ge 2$$

ii) $$x \lt 19$$

i) 

$$\underline{\{x:x\ge2\}}$$

ii)

$$\underline{\{x:x\lt19\}}$$​

Combining notations

Two different notations can be combined using a union or an intersection. 

​$$x \in A\ \cup\ B$$ is the union of $$A$$ and $$B$$. This means that $$x$$ can be in either $$A$$ or $$B$$ or both.

​$$x \in A\ \cap\ B$$ is the intersection of $$A$$ and $$B$$. This means that $$x$$ is in both $$A$$ and $$B$$.

When there are no solutions to an inequality, you can write $$x \in \empty$$, where $$\empty$$ represents the empty set.

Example 2

Write the following in set notation:

i) $$x \gt 2$$ or $$x \lt -4$$

ii) $$x \gt 5$$ and $$x \lt 3$$

iii) $$2 \le x \le 10$$

i) $$x$$ is greater than $$2$$ or $$x$$ is less than $$-4$$, therefore a union is needed:i

$$\underline{\{x:x \gt2\}\ \cup\ \{x:x\lt-4\}}$$​​

ii) $$x$$ cannot be greater than $$5$$ and less than $$3$$, therefore there are no solutions:

$$\underline{x \in \empty}$$​​

iii) $$x$$ is greater than or equal to $$2$$ and less than or equal to $$10$$, therefore an intersection is needed:

$$\underline{\{x:x \ge 2\}\ \cap\ \{x:x \le 10\}}$$