Intersection - garden

What is Intersection?

You and your friend both have playlists. You want to find songs you both have.

That’s intersection — only what’s shared.

Symbol: $\cap$ — looks like a cap (the overlap).

$A \cap B$

This reads: “A intersect B”

$A = \{1, 2, 3\}$ $B = \{3, 4, 5\}$

Find $A \cap B$ :

Check each element — is it in both sets?

$A \cap B = \{3\}$

If it’s not in both, it’s out.

$A$	$B$	$A \cap B$	Why?
$\{a, b, c\}$	$\{b, c, d\}$	$\{b, c\}$	b and c are shared
$\{1, 2\}$	$\{3, 4\}$	$\emptyset$	Nothing in common
$\{1, 2\}$	$\{1, 2, 3\}$	$\{1, 2\}$	All of A is in B
$\{1, 2, 3\}$	$\emptyset$	$\emptyset$	Empty set shares nothing

When two sets share nothing, we call them disjoint.

$A \cap B = \emptyset$

Examples: even and odd numbers, $\{1, 2\}$ and $\{3, 4\}$ .

Intersection with itself:

$A \cap A = A$

Everything in A is also in A, so you keep everything.

Intersection with empty set:

$A \cap \emptyset = \emptyset$

The empty set has nothing to share.

Order doesn’t matter (commutative):

$A \cap B = B \cap A$

“What we both have” is the same from either perspective.

Grouping doesn’t matter (associative):

$(A \cap B) \cap C = A \cap (B \cap C)$

When finding what three sets share, the grouping doesn’t matter.

$A \cap B = \{x \mid x \in A \text{ and } x \in B\}$

This reads: “The set of all $x$ such that $x$ is in A and $x$ is in B.”

Intersection asks: is it in both sets?