Lesson 10 of 10 · 15 min

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Must-know facts

18 facts

1A relation on A is a subset of A × A; with n(A) = n there are 2^(n²) of them.
2Empty relation: R = φ. Universal relation: R = A × A. Both are trivial relations.
3Reflexive: (a, a) ∈ R for all a ∈ A.
4Symmetric: (a, b) ∈ R ⇒ (b, a) ∈ R.
5Transitive: (a, b), (b, c) ∈ R ⇒ (a, c) ∈ R.
6Equivalence relation = reflexive + symmetric + transitive.
7Perpendicularity of lines: symmetric only. a ≤ b on R: reflexive and transitive, not symmetric.
8Congruence and similarity of triangles, and 'n divides a − b' on Z, are equivalence relations.
9Equivalence classes are pairwise disjoint and their union is the whole set; [a] = [b] ⇔ a R b.
102 | a − b on Z has two classes, [0] (evens) and [1] (odds); 3 | a − b has three, [0], [1], [2].
11One-one: f(x₁) = f(x₂) ⇒ x₁ = x₂. Many-one otherwise.
12Onto: range = codomain. Bijective: one-one and onto.
13f(x) = 2x: one-one not onto on N; bijective on R. x²: neither on R.
14[x], |x| and sgn x are neither one-one nor onto as maps R → R.
15For finite X, f : X → X is one-one ⇔ onto; there are n! such bijections of an n-element set.
16gof(x) = g(f(x)): f first. gof ≠ fog in general.
17f is invertible ⇔ f is one-one and onto; gof = I_X and fog = I_Y.
18f⁻¹ is found by solving y = f(x) for x; f⁻¹ ≠ 1/f.

Common traps

Where marks are lost

Calling a relation reflexive because some pairs (a, a) are present.

Every element needs its pair (a, a). On {1, 2, 3}, {(1, 1), (2, 2)} is not reflexive because (3, 3) is missing.

Declaring a relation transitive after checking one chain.

Every chain (a, b), (b, c) needs (a, c). Hunt for a failing chain; if none exists after checking all, it is transitive.

Thinking an empty relation is reflexive because it breaks no rule.

On a non-empty set the empty relation is not reflexive, since (a, a) is missing. It is symmetric and transitive, because there is no pair to test.

Assuming symmetric and transitive together force reflexive.

An element that is related to nothing never gets its (a, a). {(1, 1), (1, 2), (2, 1), (2, 2)} on {1, 2, 3} is symmetric and transitive but not reflexive.

Judging onto from the rule without looking at the codomain.

Onto means range = codomain. 2x is onto R → R but not N → N; x² is onto R → [0, ∞) but not R → R.

Reading gof(x) as f(g(x)).

The function written next to x acts first: gof(x) = g(f(x)).

Calling a one-one function invertible when its codomain is larger than its range.

An inverse needs one-one and onto. Shrink the codomain to the range, as with f(x) = 4x + 3 on N, and only then invert.

Writing f⁻¹(x) = 1/f(x).

f⁻¹ reverses the rule. For f(x) = 5x − 2, f⁻¹(x) = (x + 2)/5, not 1/(5x − 2).

Formulas

10 to know

Relations on a set

number of relations on A = 2^(n²)

n(A) = n, since A × A has n² pairs.

Reflexive

(a, a) ∈ R ∀ a ∈ A

Every diagonal pair present.

Symmetric

(a, b) ∈ R ⇒ (b, a) ∈ R

Every pair has its mirror.

Transitive

(a, b), (b, c) ∈ R ⇒ (a, c) ∈ R

Every chain has its shortcut.

Equivalence class

[a] = {x ∈ X : x R a}

Classes are equal or disjoint.

One-one

f(x₁) = f(x₂) ⇒ x₁ = x₂

Injective.

Onto

∀ y ∈ Y, ∃ x ∈ X : f(x) = y

Surjective; range = codomain.

Bijections of a finite set

n(X) = n ⇒ n! one-one functions X → X

Each is also onto.

Composition

gof(x) = g(f(x))

f : A → B, g : B → C, gof : A → C.

Inverse

f⁻¹of = I_X, fof⁻¹ = I_Y

Exists exactly when f is one-one and onto.

Key terms

12 terms

Empty relation: The relation φ on a set: no element is related to any element.
Universal relation: The relation A × A: every element is related to every element.
Reflexive relation: One in which every element is related to itself.
Symmetric relation: One in which a related to b always means b related to a.
Transitive relation: One in which a related to b and b related to c always means a related to c.
Equivalence relation: A relation that is reflexive, symmetric and transitive.
Equivalence class: [a], the set of all elements related to a under an equivalence relation.
One-one (injective): A function that never gives two different inputs the same output.
Onto (surjective): A function whose range is its whole codomain.
Bijective: A function that is both one-one and onto.
Composition: gof, the function x ↦ g(f(x)): apply f, then g.
Inverse function: f⁻¹, the function that undoes f; it exists only for a bijection.