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CA Foundation · Quantitative Aptitude

Sets, Relations and Functions, Limits and Continuity for CA Foundation

This chapter covers sets and their operations, relations, functions, limits and continuity. To solve questions, identify the definition being tested, apply a formula such as n(A ∪ B) = n(A) + n(B) − n(A ∩ B), and test options quickly. For limits, substitute first, then factorise or use standard limits.

What this chapter covers

This chapter is the base of Business Mathematics in Paper 3. It starts with sets: collections of well-defined objects, how to write them, and how to combine them with union, intersection, difference and complement. Then it moves to relations (pairs linking elements of two sets) and functions (relations where each input gets exactly one output).

The second half moves to calculus ideas. A limit tells you what value f(x) approaches as x approaches a point. Continuity asks whether the function's value at that point matches the limit. Both are tested with short, mechanical MCQs.

The chapter connects to the rest of the paper in two ways. Function notation and domain and range are used in later chapters such as differentiation and in business applications like cost and revenue. Set counting also helps in Statistics and probability, where you describe events as sets. Learn it well once and it supports several other chapters.

Paper 3 is an MCQ paper with 0.25 negative marking, so you win by being fast and accurate on questions that have a clear method. This chapter is full of those. Most questions test a definition, a formula or a short calculation, and wrong options can often be eliminated in seconds. The ideas are also needed in later Business Mathematics chapters, so effort here pays off more than once. Since the chapter is rule-based rather than lengthy, steady practice can make it one of your safer scoring areas.

Sets, Relations and Functions, Limits and Continuity: topics in the order to study them

  1. 1Sets and Their RepresentationEverything else uses set language, so learn roster form, set-builder form, and types of sets (empty, finite, subset, power set) first.
  2. 2Set Operations and Venn DiagramsOnce you know what a set is, you can combine sets and use Venn diagrams and counting formulas for word problems.
  3. 3Relations and Their TypesRelations are built from Cartesian products of sets, so you need set basics before reflexive, symmetric and transitive ideas make sense.
  4. 4Functions and Their TypesA function is a special relation, so study it after relations; then learn domain, range, one-one, onto and inverse.
  5. 5Limits and Standard Limit FormulasLimits need comfort with function notation and algebra, so they come after functions.
  6. 6Continuity of a FunctionContinuity is defined using limits, so study it last and apply the limit tests directly.

How to prepare Sets, Relations and Functions, Limits and Continuity

Spend your time on understanding definitions first, then on timed MCQ practice. The chapter is short enough to revise several times.

  1. Write the key definitions in your own words: subset, power set, reflexive, symmetric, transitive, one-one, onto, limit, continuous. Keep them on one page.
  2. Practise converting between roster form and set-builder form until it is automatic, then solve counting problems with the formula n(A ∪ B) = n(A) + n(B) − n(A ∩ B).
  3. For relations, test each property on a small example set. Check every pair, because one failing pair is enough to reject a property.
  4. For functions, find the domain first (denominator not zero, expression under an even root not negative), then check one-one and onto using simple numeric examples.
  5. For limits, always try direct substitution first. If you get 0/0, factorise and cancel, or use a standard limit. Practise each method on ten questions.
  6. For continuity, check three things at the point: f(a) exists, the limit exists, and the two are equal. For piecewise functions, compare left and right limits.
  7. Finish with a timed set of 20 mixed MCQs. Mark questions you would skip in the exam and note why, so you learn when to move on.

Common mistakes in Sets, Relations and Functions, Limits and Continuity

  • Writing repeated elements or mixing up ∈ and ⊂ in sets.

    Fix: Use ∈ for an element and ⊂ for a set inside a set. Remember {1} is a set, while 1 is an element.

  • Calling a relation symmetric or transitive after checking only a few pairs.

    Fix: List every pair and check each one. A single failure means the property does not hold.

  • Forgetting the domain restriction when working with functions.

    Fix: Before anything else, rule out zero denominators and negative values under even roots.

  • Confusing one-one with onto.

    Fix: One-one is about inputs not sharing an output. Onto is about every codomain element being reached.

  • Writing the limit as 0/0 or as undefined without simplifying.

    Fix: 0/0 is a sign to simplify. Factorise, cancel the common factor, then substitute again.

  • Declaring a piecewise function continuous by checking only f(a).

    Fix: Compute both one-sided limits and compare them with f(a) before deciding.

Last-day revision: Sets, Relations and Functions, Limits and Continuity

  • A set is a well-defined collection; the empty set has no elements and is a subset of every set.
  • A set with n elements has 2ⁿ subsets.
  • n(A ∪ B) = n(A) + n(B) − n(A ∩ B).
  • A relation from A to B is a subset of A × B; if A has m elements and B has n, then n(A × B) = mn.
  • Reflexive: (a, a) is in R for every a. Symmetric: (a, b) in R implies (b, a) in R. Transitive: (a, b) and (b, c) in R imply (a, c) in R.
  • An equivalence relation is reflexive, symmetric and transitive.
  • A function gives exactly one output for each input; no input can be left out or have two outputs.
  • One-one means different inputs give different outputs; onto means the range equals the codomain.
  • If the limits of f and g both exist (finite), then the limit of the sum, product or quotient equals the sum, product or quotient of the limits (quotient needs a non-zero denominator limit).
  • For 0/0 forms, factorise and cancel before substituting again.
  • lim (x→0) (eˣ − 1)/x = 1 and lim (x→0) log(1 + x)/x = 1.
  • f is continuous at x = a if left limit = right limit = f(a).

Sets, Relations and Functions, Limits and Continuity practice questions

Sets, Relations and Functions, Limits and Continuity: frequently asked questions

Is this chapter important for the CA Foundation Quantitative Aptitude paper?

Yes. It forms the base of Business Mathematics, and its questions are short and formula-based. That suits an MCQ paper where speed and accuracy decide your score.

Do I need to know calculus before studying limits?

No. You only need algebra: factorising, simplifying fractions and basic function notation. Limits here are taught from direct substitution and a few standard results.

How should I handle negative marking in this chapter?

Each wrong answer costs 0.25 marks. Attempt questions where you can eliminate at least two options or know the method. Skip long limit or relation-checking questions if you are unsure after a quick look.

What is the fastest way to check if a relation is an equivalence relation?

Test reflexive first, since it fails quickly if any (a, a) is missing. If it holds, check symmetric, then transitive. Stop as soon as one property fails.