Sets, Relations & Functions

Concept overview

A set is a well-defined collection of distinct objects. Standard set operations — union, intersection, difference, complement — combined with Venn diagrams and the inclusion-exclusion principle solve the count-based problems JEE loves. Cartesian product A × B has |A|·|B| ordered pairs and provides the base for relations.

A relation from A to B is a subset of A × B. Relations on a single set can be reflexive, symmetric, transitive, equivalence (all three) or partial order. Every equivalence relation partitions the set into equivalence classes.

Functions are single-valued relations. Domain, codomain and range are the three sets of a function. One-one (injective) maps distinct inputs to distinct outputs; onto (surjective) covers the entire codomain. Bijective functions (both) have inverses. Composite functions (f ∘ g)(x) = f(g(x)) chain functions together, and the domain of a composite is where both are defined and g's output lies in f's domain.

Key formulas & concepts

• Set complement: A^c = U \ A
• Inclusion-exclusion: |A ∪ B| = |A| + |B| − |A ∩ B|
• Cartesian product: |A × B| = |A| × |B|
• Number of relations: 2^(|A|·|B|) subsets of A × B
• One-one function count: n! / (n − m)! for f: A → B, |A|=m ≤ |B|=n
• Composite domain: dom(f ∘ g) = { x ∈ dom(g) : g(x) ∈ dom(f) }
• Inverse function: f exists ⇔ f is bijective; (f⁻¹)⁻¹ = f

Common mistakes to avoid

• Confusing codomain with range — codomain is the target set; range is the actual image.
• Assuming every function has an inverse — only bijective functions do.
• Ignoring the domain restriction in composite functions.
• Treating symmetric and transitive as equivalent — they are independent properties.

JEE Main relevance

JEE Main asks 1 direct sets/functions question every paper — usually on inclusion-exclusion counting or function classification (one-one, onto).

JEE Advanced relevance

Advanced problems ask for the number of onto functions between finite sets (Stirling numbers) or characterise equivalence classes on non-trivial sets.

Previous-year questions (PYQ)

Filter the PYQ bank by 'Sets and Functions'. Counting functions with given properties is the highest-frequency archetype.