Union and Intersection

The union of 2 sets $A$ and $B$ is denoted by $A \cup B$. This is the set of all distinct elements that are in $A$ or $B$. A useful way to remember the symbol is $\cup$nion. We can define the union of a collection of sets, as the set of all distinct elements that are in any of these sets.

The intersection of 2 sets $A$ and $B$ is denoted by $A \cap B$. This is the set of all distinct elements that are in both $A$ and $B$. A useful way to remember the symbol is i$\cap$tersection. We define the intersection of a collection of sets, as the set of all distinct elements that are in all of these sets.

If $A = \{ 1, 3, 5, 7, 9 \}$ and $B = \{ 2, 3, 5, 7, \}$, what are $A \cup B$ and $A \cap B$?

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We have

$$\begin{aligned} A \cup B &= \{ 1, 2, 3, 5, 7, 9 \} \\ A \cap B &= \{ 3, 5, 7 \}. \ _\square \end{aligned}$$

A great way of thinking about union and intersection is by using Venn diagrams. These are explained as follows:

We will represent sets with circles.

Then we can put the values in appropriate areas.

The Union is any region including either A or B.

The Intersection is any region including both A and B.

The diagrams we have drawn are called the Venn diagrams.