Intervals and Their Operations

25 دقيقة

الشرح

An interval is a whole stretch of the number line at once. Brackets record whether the ends are included, and set operations combine intervals.

Interval

The set of all real numbers between two endpoints. A square bracket

[\,]

includes an endpoint; a round bracket

(\,)

excludes it.

[a,b]=\{x\in\mathbb{R}:a\le x\le b\}

Infinity

Infinity is never a reachable endpoint, so

\infty

and

-\infty

always take a round bracket.

Combining intervals

Union

A\cup B

means “in either set”. Intersection

A\cap B

means “in both”. The difference and complement remove elements.

Draw it first

Sketch both intervals on one number line before you operate: a filled dot means included, an open dot means excluded.

Set up the number line.

▶ Watch an inequality become an interval on the number line.

Use the labels and colours to connect the rule with the picture.

Worked exampleInequality to interval

Write $-2<x\le5$ as an interval.

Worked exampleIntersection

Find $[-3,4)\cap(1,7]$ .

Worked exampleUnion

Find $(-\infty,2]\cup(1,6)$ .