Class 11 | CBSE | MATHEMATICS
SETS | UNION OF SETS

1.9.1 Union of Sets

Let A and B be any two sets. The union of A and B is the set which contains all elements of A and all elements of B, with common elements written only once.

The symbol ∪ is used to denote union.
Symbolically, we write A ∪ B and read it as “A union B”.

Brain Meaning:
Union means “take everything, don’t repeat”.

Example 12
Let A = { 2, 4, 6, 8 } and B = { 6, 8, 10, 12 }. Find A ∪ B.

Solution:
A ∪ B = { 2, 4, 6, 8, 10, 12 }

Neurological note:
Your brain automatically removes 6 and 8 once it sees repetition.

Example 13
Let A = { a, e, i, o, u } and B = { a, i, u }. Show that A ∪ B = A.

Solution:
A ∪ B = { a, e, i, o, u } = A

Brain Rule:
If a smaller set is fully inside a bigger set, union does not change anything.
If B ⊂ A, then A ∪ B = A

Example 14
Let X = { Ram, Geeta, Akbar } be students of Class XI in the hockey team.
Let Y = { Geeta, David, Ashok } be students of Class XI in the football team.

Solution:
X ∪ Y = { Ram, Geeta, Akbar, David, Ashok }

Interpretation:
This set represents students who play hockey or football or both.

Formal Definition:
The union of two sets A and B is the set of all elements which belong to A or B or both.

🧠 Final Brain Shortcut:
Union = OR logic + no repetition

Union of Sets — Brain-Level Meaning

Union ( ∪ ) means: collect everything without repeating.

Visually, the brain understands union as:

• Everything inside Set A
• Everything inside Set B
• Including the overlapping part
• No element is counted twice

When students see two circles with the overlap shaded, the brain instantly reads:

That is A ∪ B.

Some Properties of the Operation of Union

The operation of union ( ∪ ) follows certain fundamental laws. These laws help us simplify and understand set expressions easily.

(i) Commutative Law
A ∪ B = B ∪ A

Meaning: The order of sets does not matter. Taking elements from A and B gives the same result as taking them from B and A.

(ii) Associative Law
( A ∪ B ) ∪ C = A ∪ ( B ∪ C )

Meaning: The way sets are grouped does not affect the union. We get the same set whether we unite A and B first or B and C first.

(iii) Law of Identity Element
A ∪ φ = A

Meaning: Union with the empty set does not change the set. The empty set ( φ ) is called the identity element of union.