Class 11 | CBSE | MATHEMATICS
SETS | INTERSECTION OF SETS

1.9.2 Intersection of Sets — Brain-Level Understanding

The intersection of two sets A and B is the set of all elements that are common to both A and B. The symbol used for intersection is ∩.

Symbolic form:
A ∩ B = { x : x ∈ A and x ∈ B }

Neurological Meaning

The human brain is extremely good at detecting overlap. When two regions overlap visually, the brain instantly focuses on the shared area.

So, when students see two overlapping circles, the brain automatically reads:

“Only what is common survives.”

That shared region is exactly what intersection ( ∩ ) means.

Examples (Brain-Friendly)

Example 1:
A = {2, 4, 6, 8}, B = {6, 8, 10, 12}

A ∩ B = {6, 8}

Brain logic: Only 6 and 8 appear in both sets.

Example 2:
X = {Ram, Geeta, Akbar}
Y = {Geeta, David, Ashok}

X ∩ Y = {Geeta}

Brain logic: One name appears in both mental lists.

Example 3:
A = {1,2,3,4,5,6,7,8,9,10}
B = {2,3,5,7}

A ∩ B = {2,3,5,7} = B

Brain shortcut: When one set lies completely inside another, intersection returns the smaller set.

Disjoint Sets — Brain Alert

If two sets have no overlap, their intersection is empty.

A ∩ B = φ

Neurological cue: No overlap → nothing survives → empty set.

Example:
A = {2,4,6,8}, B = {1,3,5,7}

These sets are disjoint.

Properties of Intersection (Brain-Mapped)

(i) Commutative Law
A ∩ B = B ∩ A

Order does not matter; overlap remains the same.

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

Grouping does not change the common region.

(iii) Law of φ and U
φ ∩ A = φ ,   U ∩ A = A

Nothing intersected with something gives nothing.
Everything intersected with A gives A.

(iv) Idempotent Law
A ∩ A = A

Overlapping a set with itself changes nothing.

(v) Distributive Law
A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

The brain distributes attention: first collect (B or C), then filter what belongs to A.

Venn Diagram — Logical Intersection (A ∩ B)

🧠 Neurological rule:
The brain trusts geometry more than labels.
Red appears only where both sets exist simultaneously — that visual truth is the meaning of intersection.