PYQ Vault

Playbook

Relations and Functions

Grown into a cornerstone. Relations are careful counting, often with a numeric answer; functions are domain, range, counting maps and functional equations.

Questions in the bank
199
q/paper in 2025–26
1.73
Numeric answer
24%
Notes pages
8

Tier: Cornerstone

When you’ll see it

A relation on a finite set, functions between finite sets to count, or a formula whose domain, range, inverse or functional rule is asked.

How this chapter is tested

The chapter splits into two halves. The relations half — sets, the three properties and counting pairs — is careful counting and is often set as a numeric-answer question, with no options to check against. The functions half is domain, range, counting maps, composition and functional equations.

Domain questions usually ask for a sum of interval endpoints, so one missed excluded point changes the answer. The conditions stack: a square root needs its argument ≥ 0, a log needs > 0, a log in a denominator also needs its argument ≠ 1, and sin⁻¹ and cos⁻¹ need −1 to 1.

Counting functions is permutations and combinations in disguise: nᵐ maps from an m-set to an n-set, n!/(n − m)! one-one maps, onto maps by inclusion–exclusion. Functional equations reward substitution — x = y = 0, then y = −x or x → 1/x — rather than guessing a formula.

The sub-skills

The distinct skills inside the chapter, in the order to learn them.

  • Sets and inclusion–exclusion

    n(A ∪ B) = n(A) + n(B) − n(A ∩ B); a set with n elements has 2ⁿ subsets.

  • Reflexive, symmetric, transitive

    Prove a property with a general argument; break one with a single counterexample.

  • Counting relations

    Count ordered pairs; on n elements there are 2^(n² − n) reflexive relations and 2^(n(n + 1)/2) symmetric ones.

  • Domain

    List the conditions from roots, logs, denominators and inverse trigonometric functions, then intersect them.

  • Range, one-one and onto

    Bound the function, use the discriminant in x, or use monotonicity; onto means the range equals the codomain.

  • Counting functions

    nᵐ maps, n!/(n − m)! one-one maps, onto maps by inclusion–exclusion.

  • Composition, inverses and functional equations

    f∘g applies g first; iterate until f repeats; solve a rule such as f(x + y) = f(x) + f(y) by substituting convenient values.

Traps to expect

Distractor shapes this chapter reuses. The Traps page covers the ones that cut across chapters.

  • A log in a denominator

    1/log u needs u > 0 and u ≠ 1. The missing point is the usual gap between two options.

  • Reflexive and symmetric but not transitive

    |a − b| ≤ 1 relates 1 to 2 and 2 to 3 but not 1 to 3. Test a chain that crosses the limit before calling a relation an equivalence.

  • Onto depends on the codomain

    The same formula can be onto one codomain and not another. Compare the range with the stated codomain.

  • Greatest integer back to intervals

    [x] ≤ −3 means x < −2, not x ≤ −3: every x with integer part −3 lies in [−3, −2).

Learn it before you drill it

This chapter has full teaching notes — foundations, worked examples, self-checks and a mastery check for each page. Read the notes once, then drill page by page below.

Relations and Functions notes

Drill every Relations and Functions question

199 questions from the bank, across 8 subtopics.

Drill the 199 questions

Drill one subtopic at a time

The 8 subtopics, in teaching order.

Related playbooks

Often paired with this one — the technique or the trap overlaps. Drill these next.