Union - garden

What is Union?

You have two playlists. You want to combine them into one.

That’s union — everything from both sets, no duplicates.

Symbol: $\cup$ — looks like a cup that holds everything.

$A \cup B$

This reads: “A union B”

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

Find $A \cup B$ :

Start with A, then add anything from B that’s not already there:

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

Duplicates are ignored. The 3 appears once.

$A$	$B$	$A \cup B$	Why?
$\{a, b\}$	$\{c, d\}$	$\{a, b, c, d\}$	No overlap, just combine
$\{1, 2, 3\}$	$\{1, 2, 3\}$	$\{1, 2, 3\}$	Identical sets
$\{1, 2\}$	$\{1, 2, 3, 4\}$	$\{1, 2, 3, 4\}$	A inside B, union is B
$\{1, 2, 3\}$	$\emptyset$	$\{1, 2, 3\}$	Empty set adds nothing

Union with itself:

$A \cup A = A$

Combining something with itself doesn’t add anything new.

Union with empty set:

$A \cup \emptyset = A$

Adding nothing changes nothing.

Order doesn’t matter (commutative):

$A \cup B = B \cup A$

Combining A with B is the same as combining B with A.

Grouping doesn’t matter (associative):

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

When combining three sets, the grouping doesn’t affect the result.

$A \cup B = \{x \mid x \in A \text{ or } x \in B\}$

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

In math, “or” includes “both” — if something is in both sets, it’s included.

Union asks: is it in either set?