5.1 Number Systems, Sets & Functions
Key Takeaways
- Real numbers nest as ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ; complex numbers ℂ extend ℝ with i where i² = −1.
- A function assigns each domain element exactly one range value; one-to-one and onto decide invertibility.
- Set operations ∪, ∩, − and complements obey De Morgan: (A ∪ B)′ = A′ ∩ B′ and (A ∩ B)′ = A′ ∪ B′.
- Composite (f ∘ g)(x) = f(g(x)); evaluate the inner function first on timed MCQs.
- PAF CAE maths papers are commonly reported as ~50 MCQs in ~25 minutes—expect fast set/function identification, not long proofs.
Why This Matters for the PAF CAE Academic Paper
Mathematics is commonly weighted around 30% of the academic portion for Pakistan Air Force College of Aeronautical Engineering (CAE) and related engineering-branch initial tests. Coaching and candidate reports often describe a computer-based maths paper of roughly 50 MCQs in about 25 minutes (confirm your cycle’s slip on joinpaf.gov.pk—exact counts are not always published the same way every year). That timing leaves about 30 seconds per question, so algebra items reward instant recognition of number types, set identities, and function properties rather than lengthy written solutions.
This section covers the FSc Pre-Engineering / A-Level foundation: number systems, sets, relations, and functions. Later sections build equations, sequences, and logarithms on this base.
Number Systems (Nesting You Must Recall Instantly)
| Symbol | Name | Typical FSc description | Example |
|---|---|---|---|
| ℕ | Natural numbers | Counting numbers (texts vary on whether 0 ∈ ℕ) | 1, 2, 3, … |
| ℤ | Integers | Whole numbers and their negatives | …, −2, −1, 0, 1, 2 |
| ℚ | Rational numbers | Ratios p/q with p, q ∈ ℤ, q ≠ 0 | 3/4, −7, 0.125 |
| ℝ | Real numbers | All rationals and irrationals | √2, π, −5 |
| ℂ | Complex numbers | a + bi with a, b ∈ ℝ and i² = −1 | 3 − 2i |
Nesting (standard exam order): ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ ⊂ ℂ.
Worked classification. Classify √4, √2, 22/7, and 3 + 4i.
- √4 = 2, which is natural, integer, rational, and real.
- √2 is irrational (not p/q), hence real but not rational.
- 22/7 is rational (exact fraction)—do not confuse it with π, which is irrational.
- 3 + 4i is complex and non-real if you treat the imaginary part as nonzero.
Complex arithmetic (exam-speed). (2 − 3i)(1 + 4i) = 2·1 + 2·4i − 3i·1 − 3i·4i = 2 + 8i − 3i − 12i². Since i² = −1, −12i² = 12, so the product is 14 + 5i.
Modulus: |a + bi| = √(a² + b²). For 3 − 4i, |z| = √(9 + 16) = 5. Expect at least one modulus or conjugate item mixed into algebra banks.
Sets: Language and Operations
A set is a well-defined collection of objects. Use roster form {1, 2, 3} or set-builder {x ∈ ℝ : x² < 4}.
| Operation | Meaning | Symbol |
|---|---|---|
| Union | Elements in A or B (or both) | A ∪ B |
| Intersection | Elements in both A and B | A ∩ B |
| Difference | In A but not in B | A − B or A \ B |
| Complement | In universal set U but not in A | A′ or Aᶜ |
| Cartesian product | Ordered pairs (a, b) | A × B |
De Morgan’s laws (memorize):
- (A ∪ B)′ = A′ ∩ B′
- (A ∩ B)′ = A′ ∪ B′
Worked Venn count. In a batch of 40 CAE aspirants, 25 practise Physics daily, 22 practise Maths daily, and 12 practise both. How many practise at least one of the two?
|P ∪ M| = |P| + |M| − |P ∩ M| = 25 + 22 − 12 = 35. Then 40 − 35 = 5 practise neither. Only Physics: 25 − 12 = 13. Only Maths: 22 − 12 = 10.
Power set. If A = {a, b}, then P(A) = {∅, {a}, {b}, {a, b}} and |P(A)| = 2^{|A|} = 4. Questions often ask |P(A)| when |A| = 3 → answer 8.
Relations vs Functions
A relation from A to B is any subset of A × B. A function f: A → B assigns to each x ∈ A exactly one y ∈ B.
Vertical-line test (graphs): a curve is a function of x if no vertical line meets it more than once.
Types tested on MCQs:
- One-to-one (injective): f(x₁) = f(x₂) ⇒ x₁ = x₂. Horizontal-line test: no horizontal line meets the graph more than once.
- Onto (surjective): range equals codomain.
- Bijective: both one-to-one and onto → inverse function exists.
Worked function check. Is f: ℝ → ℝ, f(x) = x² a function? Yes—each x has one square. Is it one-to-one on ℝ? No: f(2) = f(−2) = 4. Is it onto ℝ? No: −1 has no real square root. Restrict domain to [0, ∞) and codomain to [0, ∞) and it becomes bijective with inverse √x.
Domain, Range, and Composition
Domain = allowed inputs; range = actual outputs.
Examples at FSc speed:
- f(x) = 1/(x − 3) → domain ℝ \ {3}.
- g(x) = √(x − 2) → domain [2, ∞), range [0, ∞).
- h(x) = ln(x + 1) → domain (−1, ∞) (natural log defined for positive arguments).
Composition: (f ∘ g)(x) = f(g(x)). Always evaluate g first.
Worked composition. Let f(x) = 2x + 1 and g(x) = x² − 3. Then (f ∘ g)(2) = f(g(2)) = f(4 − 3) = f(1) = 3. And (g ∘ f)(2) = g(5) = 25 − 3 = 22. Order matters—common trap.
Inverse idea. If f(x) = 3x − 5, set y = 3x − 5 → x = (y + 5)/3 → f⁻¹(x) = (x + 5)/3. Check: f(f⁻¹(x)) = x.
Even and Odd Functions (Quick Discriminators)
- Even: f(−x) = f(x) (symmetry about y-axis)—e.g. x², cos x.
- Odd: f(−x) = −f(x) (origin symmetry)—e.g. x³, sin x.
- Neither: most mixed polynomials like x² + x.
Worked test. f(x) = x³ − 4x. f(−x) = −x³ + 4x = −(x³ − 4x) = −f(x) → odd.
On the Exam
Expect rapid items: “Which is irrational?”, “|A ∪ B| given …”, “Is f one-to-one?”, “(f ∘ g)(a) = ?”, and occasional |a + bi|. Under a commonly reported ~25-minute maths window, skip multi-step set-word puzzles if the algebra stem is clearer—return only if time remains.
If A = {1, 2, 3} and B = {2, 3, 4}, what is |A ∪ B|?
For f(x) = 2x + 1 and g(x) = x² − 3, what is (f ∘ g)(2)?
Which statement about f: ℝ → ℝ given by f(x) = x² is correct?
What is |(3 − 4i)|?