Sets & Venn DiagramsFoundation

Union, intersection, complement — interactive Mathematics simulation for IIT-JEE.

Concept

A Venn diagram shows how two sets overlap inside a universal set. The key counting tool is the inclusion–exclusion principle, which avoids double-counting the shared region $A\cap B$ .

Key formula

n(A\cup B) = n(A) + n(B) - n(A\cap B)

Derivation

If you simply add $n(A)$ and $n(B)$ , every element in the overlap is counted twice.

Subtracting $n(A\cap B)$ once corrects this, giving the size of the union. Whatever is left in the universal set, $n(U)-n(A\cup B)$ , belongs to neither set.

Scenarios to explore

Sets & Venn Diagrams — Union, intersection, complement & inclusion–exclusion.

Real-world applications

Survey problems ('how many study both subjects?').
Probability of combined events.
Database queries combining filters.

JEE exam tips

'Only A' = n(A) − n(A ∩ B); 'exactly one' = n(A) + n(B) − 2·n(A ∩ B).
n(A′) = n(U) − n(A) by the complement rule.
De Morgan: (A ∪ B)′ = A′ ∩ B′.

Common mistakes

Adding n(A) and n(B) without subtracting the overlap.
Confusing 'A only' with n(A) — A also contains the intersection.
Letting n(A ∩ B) exceed n(A) or n(B).

Exam traps to avoid

'At least one' means the union; 'exactly one' excludes the intersection twice over.
Three-set problems need the full inclusion–exclusion with the triple overlap added back.