🎯 Key Points
- Ordered pair (a, b) ≠ (b, a); Cartesian product A × B = {(a,b) : a∈A, b∈B}; n(A×B) = n(A)·n(B)
- Relation R from A to B: any subset of A×B; domain = set of first elements; range = set of second elements
- Function f: A→B assigns exactly one element of B to each element of A; domain=A, codomain=B, range⊆B
- One-one (injective): distinct inputs → distinct outputs; Onto (surjective): range = codomain; Bijective: both one-one and onto
- Vertical line test: graph represents a function if every vertical line meets it at most once
- Composite function (f∘g)(x) = f(g(x)); domain of f∘g = {x ∈ domain(g) : g(x) ∈ domain(f)}
- Inverse function f⁻¹ exists only if f is bijective; (f⁻¹∘f)(x) = x
- Even function: f(−x) = f(x); Odd function: f(−x) = −f(x)
Types of Functions
- Identity: f(x) = x for all x; Constant: f(x) = c
- Modulus: f(x) = |x|; domain ℝ, range [0,∞); V-shaped graph with vertex at origin
- Signum: sgn(x) = −1 if x<0, 0 if x=0, 1 if x>0; range = {−1,0,1}
- Greatest Integer (floor): ⌊x⌋ = greatest integer ≤ x; e.g. ⌊3.7⌋ = 3, ⌊−2.3⌋ = −3; step graph
- Polynomial: f(x) = aₙxⁿ + … + a₀; Rational: f(x) = p(x)/q(x), q(x)≠0
Worked Example: Composite Function
Let f(x) = 2x + 1 and g(x) = x². Find (f∘g)(3) and (g∘f)(3).
(f∘g)(3) = f(g(3)) = f(9) = 2(9) + 1 = 19
(g∘f)(3) = g(f(3)) = g(7) = 7² = 49 — note f∘g ≠ g∘f in general.
Worked Example: Domain and Range
Find domain of f(x) = √(x − 2) / (x − 5).
Need x − 2 ≥ 0 (square root) → x ≥ 2; and x − 5 ≠ 0 → x ≠ 5.
Domain = [2, 5) ∪ (5, ∞)
Cartesian Product: Properties
- A × B = {(a, b) : a ∈ A, b ∈ B}; in general A × B ≠ B × A unless A = B or one set is empty.
- n(A × B) = n(A) · n(B); if n(A) = p and n(B) = q, the number of relations from A to B is 2^(pq) (each of the pq ordered pairs is either in or out).
- A × (B ∪ C) = (A × B) ∪ (A × C) and A × (B ∩ C) = (A × B) ∩ (A × C).
- If A × B has a pair (x, y), then x is a first coordinate and y a second; ordered pair equality: (a, b) = (c, d) ⇔ a = c and b = d.
Types of Relations
A relation R on a set A (R ⊆ A × A) may be:
- Reflexive: (a, a) ∈ R for all a ∈ A.
- Symmetric: (a, b) ∈ R ⇒ (b, a) ∈ R.
- Transitive: (a, b) ∈ R and (b, c) ∈ R ⇒ (a, c) ∈ R.
- Equivalence relation: reflexive, symmetric AND transitive together (e.g. "≡ mod n", "is congruent to", "is parallel to").
- Empty relation (R = ∅) and universal relation (R = A × A) are the two extreme cases.
Domain and Range of Standard Functions
| Function | Domain | Range |
|---|---|---|
| f(x) = x² | R | [0, ∞) |
| f(x) = √x | [0, ∞) | [0, ∞) |
| f(x) = 1/x | R − {0} | R − {0} |
| f(x) = |x| | R | [0, ∞) |
| f(x) = ⌊x⌋ | R | Z (integers) |
| f(x) = 1/√x | (0, ∞) | (0, ∞) |
Algebra of Real-Valued Functions
- (f ± g)(x) = f(x) ± g(x); (f·g)(x) = f(x)·g(x); (f/g)(x) = f(x)/g(x) where g(x) ≠ 0.
- The domain of f ± g and f·g is the intersection domain(f) ∩ domain(g).
- The domain of f/g is domain(f) ∩ domain(g) with all points where g(x) = 0 removed.
- (kf)(x) = k·f(x) for a scalar k; these operations always shrink (never expand) the common domain.
Composition of Functions: Properties
- For f: A→B and g: B→C, the composite g∘f: A→C is defined by (g∘f)(x) = g(f(x)); it requires range(f) ⊆ domain(g).
- Not commutative: in general f∘g ≠ g∘f.
- Associative: h∘(g∘f) = (h∘g)∘f whenever the composites are defined.
- Composition preserves type: if f and g are both one-one, g∘f is one-one; if both onto, g∘f is onto; hence composite of two bijections is a bijection.
- Identity: f∘IA = f and IB∘f = f, where I denotes the identity function.
Invertible Functions and Inverse
- A function f: A→B is invertible if and only if it is a bijection (one-one and onto).
- The inverse f-1: B→A satisfies f-1∘f = IA and f∘f-1 = IB; it is unique.
- To find f-1: write y = f(x), solve for x in terms of y, then swap to get f-1(y).
- Inverse of a composite (reversal law): (g∘f)-1 = f-1∘g-1.
- Example: f(x) = 2x + 3 on ℝ is a bijection, with f-1(x) = (x − 3)/2.
Binary Operations
A binary operation ∗ on a set A is a rule that assigns to each ordered pair (a, b) of A × A a unique element a ∗ b of A (closure). Standard examples: + and × on ℝ.
- Commutative: a ∗ b = b ∗ a for all a, b ∈ A.
- Associative: (a ∗ b) ∗ c = a ∗ (b ∗ c) for all a, b, c ∈ A.
- Identity element e: a ∗ e = e ∗ a = a for all a; e.g. 0 for +, 1 for ×.
- Inverse of a: element b with a ∗ b = b ∗ a = e; e.g. −a for +, 1/a (a≠0) for ×.
- An operation table (Cayley table) can be used to test commutativity (symmetry about the diagonal) and read off identity and inverses.
Equivalence Classes and Partitions
- An equivalence relation R on A splits A into disjoint equivalence classes: [a] = {x ∈ A : (x, a) ∈ R}.
- Any two classes are either identical or disjoint, and their union is all of A — so R produces a partition of A.
- Conversely, every partition of A defines an equivalence relation whose classes are the parts.
- Example: "congruent modulo 3" on ℤ gives three classes — remainders 0, 1 and 2.