Sets, Relations and Functions; Limits and Continuity for CA Foundation
Sets and their operations, Venn diagrams, Cartesian products, relations and functions, and the basic ideas of limits and continuity.
Quantitative AptitudePaper 3, Business Mathematics11 MCQs2 lessons2 flashcards3 memory tricks
Practise Sets, Relations and Functions; Limits and Continuity free
What you'll learn
- Sets, relations and functions
Use set operations and classify relations and functions. - Limits and continuity
Evaluate simple limits and test continuity.
Key formulas
| Number of subsets | 2ⁿ |
|---|---|
| Union count | n(A ∪ B) = n(A) + n(B) − n(A ∩ B) |
| Cartesian product | n(A × B) = n(A) × n(B) |
| Number of relations A → B | 2^(n(A) × n(B)) |
| Complement | n(A′) = n(U) − n(A) |
| Algebraic | lim(x→a) (xⁿ − aⁿ)/(x − a) = naⁿ⁻¹ |
| Exponential | lim(x→0) (eˣ − 1)/x = 1; (aˣ − 1)/x = log a |
| e | lim(n→∞) (1 + 1/n)ⁿ = e |
Concepts explained
Sets and their operations
A set is a well-defined collection; union, intersection, complement and Cartesian product combine sets.
What it means
- n subsets: 2ⁿ (proper subsets 2ⁿ − 1).
- n(A ∪ B) = n(A) + n(B) − n(A ∩ B).
- n(A × B) = n(A) × n(B).
- n(A′) = n(U) − n(A); De Morgan: (A ∪ B)′ = A′ ∩ B′.
- 20 + 15 − 5
Answer: 30
- 50 students; 45 in E ∪ L
Answer: 5
Related terms
- Venn diagram: picture of sets as circles
- Power set: set of all subsets
Exam tip: For 'neither' questions, find the union first and subtract from the total.
Relations and functions
A relation is any subset of A × B; a function is a relation in which every element of the domain has exactly one image.
What it means
- Number of relations from A to B: 2^(n(A)·n(B)).
- Equivalence relation: reflexive, symmetric, transitive.
- One-one: distinct inputs give distinct outputs; onto: range = co-domain.
- Composite function: (f∘g)(x) = f(g(x)).
- f(x) = 2x + 3
Answer: f(2) = 7
- f(x) = x²
Answer: f(2) = f(−2) = 4
Related terms
- Domain and range: inputs allowed and outputs produced
- Inverse function: exists only for one-one onto functions
Exam tip: For one-one, check if two different inputs can give the same output.
Limits and continuity
A limit is the value a function approaches as x approaches a point. A function is continuous at a if the limit equals the function's value there.
What it means
- 0/0: factorise and cancel, or use a standard limit.
- (xⁿ − aⁿ)/(x − a) → naⁿ⁻¹.
- (eˣ − 1)/x → 1; (aˣ − 1)/x → log a; (1 + 1/n)ⁿ → e.
- Continuity: left limit = right limit = f(a).
- (x² − 4)/(x − 2) as x → 2
Answer: 4
- (x³ − 27)/(x − 3) as x → 3
Answer: 27
Related terms
- Derivative: limit of the difference quotient
- Discontinuity: a jump or gap in the graph
Exam tip: Always try direct substitution first; only if it gives 0/0 do you need algebra.
Memory tricks
Number of subsets of an n-element set = 2ⁿ.
Equivalence relation = reflexive + symmetric + transitive.
Continuous at a ⇔ lim f(x) = f(a).
Try 3 CA Foundation-style questions
Q1. The number of subsets of the set A = {1, 2, 3} is
- 6
- 8
- 3
- 9
Show answer
Answer: (2) 8
A set with n elements has 2ⁿ subsets: 2³ = 8 (including ∅ and A itself).
Q2. If f(x) = 2x + 3, then f(2) is
- 5
- 10
- 4
- 7
Show answer
Answer: (4) 7
Substitute x = 2: 2(2) + 3 = 7.
Q3. A relation that is reflexive, symmetric and transitive is called
- a function
- an identity relation
- an equivalence relation
- an empty relation
Show answer
Answer: (3) an equivalence relation
Equivalence relation = reflexive + symmetric + transitive, e.g. 'is congruent to' or 'has the same remainder as'.
Practise all 11 questions Free · explanations for every option · see where you go wrong