Union vs. Intersection: Reading Sets in Interval Notation

Union means “or”; intersection means “and”

The union $A\cup B$ contains values that are in $A$, in $B$, or in both. The intersection $A\cap B$ contains only values that satisfy both sets at the same time. Reading union as “or” and intersection as “and” makes the logical difference concrete.

Translate each set before combining it

Convert each inequality or set-builder statement to interval notation first. A strict inequality uses a parenthesis at its finite endpoint; an inclusive inequality uses a bracket. Infinity and negative infinity always use parentheses because they are directions, not attainable endpoints.

Once both intervals are on a number line:

• shade every point covered by either interval to find the union;
• keep only the overlapping shading to find the intersection.

A fresh example

Let $C=\{x\mid x<-2\}$ and $D=\{x\mid x\ge4\}$. Then

$$C=(-\infty,-2),\qquad D=[4,\infty).$$

Their union is

$$(-\infty,-2)\cup[4,\infty),$$

while their intersection is $\varnothing$ because no real number can satisfy both conditions.

Audit the endpoints last

After finding the coverage, return to every finite endpoint. Use a bracket only when that exact endpoint belongs to the resulting set. This final endpoint check prevents an otherwise correct union or intersection from being marked wrong because of one open or closed boundary.