Sets, Relations & Functions

Sets, relations, and functions form the mathematical language upon which all of JEE mathematics is built. A set is a well-defined collection of distinct objects. The key operations — union (A U B), intersection (A ∩ B), complement (A'), difference (A - B), and symmetric difference (A $\Delta$ B) — follow De Morgan's Laws: (A U B)' = A' ∩ B' and (A ∩ B)' = A' U B'. For finite sets, the inclusion-exclusion principle states |A U B| = |A| + |B| - |A ∩ B|, extending to three sets as |A U B U C| = |A| + |B| + |C| - |A ∩ B| - |B ∩ C| - |A ∩ C| + |A ∩ B ∩ C|.

A relation R from set A to set B is a subset of the Cartesian product A x B. Relations on a set A can possess four key properties: reflexive (aRa for all a), symmetric (aRb implies bRa), antisymmetric (aRb and bRa implies a = b), and transitive (aRb and bRc implies aRc). An equivalence relation satisfies reflexive, symmetric, and transitive properties simultaneously, partitioning the set into disjoint equivalence classes. A partial order satisfies reflexive, antisymmetric, and transitive properties.

A function f: A → B is a special relation where every element of A maps to exactly one element of B. A is the domain, B is the codomain, and f(A) is the range. Functions are classified as: one-to-one (injective) if f(a1) = f(a2) implies a1 = a2; onto (surjective) if range equals codomain; and bijective if both injective and surjective. The number of functions from a set of m elements to n elements is n^m; the number of onto functions uses the inclusion-exclusion formula; the number of bijections is n! (when m = n).

Composition of functions (fog)(x) = f(g(x)) is associative but not commutative. The inverse function f^(-1) exists only when f is bijective, and satisfies f^(-1)(f(x)) = x and f(f^(-1)(y)) = y. For finding the inverse, replace f(x) = y, solve for x in terms of y, then swap variables.

Even functions satisfy f(-x) = f(x) (symmetric about y-axis); odd functions satisfy f(-x) = -f(x) (symmetric about origin). Every function can be uniquely decomposed as the sum of an even function and an odd function: f(x) = [f(x)+f(-x)]/2 + [f(x)-f(-x)]/2.

The key problem-solving concept is determining the nature of a relation (equivalence vs. partial order) or function (injective, surjective, bijective) by systematically checking each defining property.