Relations and Functions
This domain tests whether you can classify a relation on a set, decide when a function is one-one or onto, and handle composites and inverses. BSEH papers usually set a 1-mark definition check, a 2–3 mark proof that a relation is an equivalence relation, and a 3–4 mark inverse-function question.
Core definitions
- R on A is reflexive if (a, a) ∈ R for every a ∈ A; symmetric if (a, b) ∈ R forces (b, a) ∈ R; transitive if (a, b), (b, c) ∈ R force (a, c) ∈ R. All three together make R an equivalence relation.
- The number of relations from a set with m elements to a set with n elements is 2^(mn).
- f: A → B is one-one if f(x₁) = f(x₂) ⇒ x₁ = x₂, and onto if every element of B is an image of some element of A.
Inverses
A function is invertible exactly when it is bijective. To find f⁻¹, write y = f(x), solve for x in terms of y, then interchange the letters.
Worked example
Let f: R → R, f(x) = (3x + 2)/5. If f(x₁) = f(x₂), then (3x₁ + 2)/5 = (3x₂ + 2)/5, so x₁ = x₂ and f is one-one. Given any y ∈ R, y = (3x + 2)/5 gives x = (5y − 2)/3, which is always real, so f is onto. Hence f⁻¹(y) = (5y − 2)/3.
Traps
- Assuming a function is onto because its outputs look 'all real'; check the stated codomain, not the range.
- Writing f⁻¹(x) = 1/f(x).
- For an equivalence proof, forgetting to show reflexivity separately from symmetry and transitivity.