Sets, Relations and Functions
Sets, Relations and Functions underpins every later chapter, so the paper tests it both directly and inside other questions. Expect set operations, counting of subsets and Cartesian products, classification of relations by their properties, and decisions about whether a function is one-one, onto or invertible.
Set rules to have ready
- For finite sets, n(A∪B) = n(A) + n(B) − n(A∩B); for three sets, subtract all three pairwise intersections and add the triple intersection back.
- De Morgan: (A∪B)′ = A′∩B′ and (A∩B)′ = A′∪B′.
- A set with n elements has 2ⁿ subsets and 2ⁿ − 1 proper subsets. If |A| = m and |B| = n, then A×B has mn elements.
- A∩(B∪C) = (A∩B)∪(A∩C).
Relations and functions
A relation on A is any subset of A×A. It is reflexive if (a,a) ∈ R for every a, symmetric if (a,b) ∈ R forces (b,a) ∈ R, and transitive if (a,b) and (b,c) in R force (a,c) in R. An equivalence relation has all three. A function f: A→B gives exactly one image to every element of A; the exam frequently tests one-one (distinct inputs, distinct outputs) and onto (every element of B is hit). Composition works right to left: (g∘f)(x) = g(f(x)). An inverse exists only for a bijection.
Worked example
Find the inverse of f(x) = (2x+3)/5. Put y = (2x+3)/5, so 5y = 2x+3 and x = (5y−3)/2. Interchanging names gives f⁻¹(x) = (5x−3)/2. Test with x = 1: f(1) = 1 and f⁻¹(1) = 1, as required.
Traps
- Calling a relation a function when one input has two images.
- Reversing the order in (g∘f).
- Leaving the empty set and the whole set out of a subset count.
- Assuming every function has an inverse.