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.
- Practise fast set and function identification only as a self-chosen drill; current public PAF sources do not publish a mathematics item count or timing for this route.
Why Numbers, Sets, and Functions Matter
Numbers, sets, and functions are foundational school-level mathematics for technical study. This independent review emphasises recognition of number types, set identities, and function properties. Current public PAF sources do not publish a mathematics weight, item count, timing, or confirmed subject allocation for this route.
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 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.
Compact complex-arithmetic example. (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 students, 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.
Useful classification types:
- 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.
Practice Focus
Useful self-checks include “Which is irrational?”, “|A ∪ B| given …”, “Is f one-to-one?”, “(f ∘ g)(a) = ?”, and occasional |a + bi|. Practise them first for accuracy, then reduce your self-imposed time without treating that drill pace as an official PAF limit.
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)|?