NCERT Solutions for Class 12 Mathematics Chapter 1 Relations and Functions

Short Note:

• A relation R on a set A is reflexive if (a, a) ∈ R for every a ∈ A; symmetric if (a, b) ∈ R implies (b, a) ∈ R; and transitive if (a, b), (b, c) ∈ R implies (a, c) ∈ R.

• An equivalence relation satisfies all three properties — reflexive, symmetric, and transitive — simultaneously, and it partitions the set into disjoint equivalence classes.

Short Note:

• A function f: A → B is one-one (injective) if distinct elements of A always map to distinct elements of B; it is onto (surjective) if every element of B has at least one preimage in A.

• A function that is both one-one and onto is called bijective, and only bijective functions possess a well-defined inverse function.

Short Note:

• The composition (gof)(x) = g(f(x)) applies the function f first, followed by g; composition of functions is associative but generally not commutative.

• A function f is invertible if and only if it is bijective, and its inverse f⁻¹ satisfies f⁻¹(f(x)) = x and f(f⁻¹(y)) = y.