Set union and intersection

Given two sets, we often want to combine them. The union collects elements from both.

To form a union, we collect every element from either set. When the same element appears in both sets, we list it just once - sets don't have duplicates.

Let $A = \{1, 2, 3\}$ and $B = \{2, 4, 6\}$. Find $A \cup B$.

Solution

The union $A \cup B$ contains all elements that are in $A$, in $B$, or in both.

From $A$: $1, 2, 3$

From $B$: $2, 4, 6$

Combining these (noting that $2$ appears in both, but we list it only once):

Sometimes we want elements that satisfy both conditions simultaneously. This is the intersection.

The key word is "and". To form an intersection, we keep only elements that appear in both sets - everything else is excluded.

For example, the set of odd integers and the set of even integers are disjoint - no integer is both odd and even, so their intersection is $\emptyset$.

Let $A = \{1, 2, 3\}$ and $B = \{2, 4, 6\}$.

Find $A \cap B$. Then let $C = \{5, 6\}$ and find $B \cap C$.

Solution

Finding $A \cap B$:

The intersection contains only elements in both sets. Checking which elements of $A$ are also in $B$:

• $1$: not in $B$
• $2$: in $B$
• $3$: not in $B$

Finding $B \cap C$:

Checking which elements of $B = \{2, 4, 6\}$ are also in $C = \{5, 6\}$:

• $2$: not in $C$
• $4$: not in $C$
• $6$: in $C$

A Venn diagram is a visual way to represent sets and their relationships. The outer rectangle represents all elements under consideration - we label it $U$. Sets are drawn as overlapping circles inside.

The overlapping circles create four distinct regions:

• $A$ only (left crescent): elements in $A$ but not $B$
• $B$ only (right crescent): elements in $B$ but not $A$
• Overlap ($A \cap B$): elements in both sets
• Outside both: elements in neither set

Each element belongs to exactly one region. Use the buttons below to see how the union and intersection appear on a Venn diagram.

The union $A \cup B$ covers all three inner regions (both circles). The intersection $A \cap B$ covers only the overlap.

Draw a Venn diagram for $U = \{1, 2, 3, 4, 5, 6\}$ with $A = \{1, 2, 3\}$ and $B = \{2, 4, 6\}$. Place each element in the correct region. Identify $A \cup B$ and $A \cap B$ from the diagram.

Solution

First, we classify each element by which sets it belongs to:

• $1$: in $A$ only
• $2$: in both $A$ and $B$
• $3$: in $A$ only
• $4$: in $B$ only
• $5$: in neither
• $6$: in $B$ only

Reading from the diagram:

Union $A \cup B$: all elements inside either circle - $\{1, 2, 3, 4, 6\}$

Intersection $A \cap B$: only the element in the overlap - $\{2\}$