Introduction
This chapter covers Relations and Functions in a compact, revision-friendly format: precise definitions, important theorems with proofs, formulas, diagrams, solved examples, a summary table and practice questions so the whole chapter sticks after one strong study session.
1. Relation — Definition & Types
1.1 Definition
Given sets A and B, the Cartesian product A × B is the set of all ordered pairs (a, b). A relation R from A to B is any subset R ⊆ A × B. If (a, b) ∈ R we write aRb.
1.2 Properties of relations
Definitions:
- Reflexive: ∀a ∈ A, (a,a) ∈ R.
- Symmetric: If (a,b) ∈ R then (b,a) ∈ R.
- Transitive: If (a,b) ∈ R and (b,c) ∈ R then (a,c) ∈ R.
1.3 Special relations
- Empty relation, universal relation, identity relation
- Equivalence relation: A relation that is reflexive, symmetric and transitive. Equivalence relations partition the set into equivalence classes.
Mapping diagram (visual)
Use a mapping diagram for clarity: draw two columns (Domain and Codomain), draw arrows to show ordered pairs. Example images/diagrams help memory a lot — add mapping diagrams as PNG/SVG files here:
2. Function — Definition, Types & Properties
2.1 Definition
A function f from A to B (written f: A → B) is a relation that assigns to each element a ∈ A exactly one element b ∈ B. That is, for every a ∈ A there exists a unique b such that (a, b) ∈ f.
2.2 Types of functions
- Injective (one-to-one): distinct inputs map to distinct outputs.
- Surjective (onto): range = codomain; every element of codomain has a preimage.
- Bijective: both injective and surjective (invertible).
- Other examples: constant function, identity function, many-to-one.
2.3 Composition and inverse
If f: A → B and g: B → C then the composition g ∘ f: A → C is defined by (g ∘ f)(a) = g(f(a)). If f is bijective then inverse f⁻¹: B → A exists with f⁻¹(f(a)) = a.
2.4 Domain, Codomain, Range (Image)
Domain — set of inputs. Codomain — set where outputs live. Range/Image — actual outputs attained by the function (subset of codomain).
3. Important Theorems & Formulas
Counting relations & functions
If |A| = m and |B| = n, total possible relations from A to B = 2^(m·n). Number of possible functions from A to B = n^m (each of the m domain elements choose one of n codomain elements).
4. Solved Examples
Example 1: A = {1,2,3}, B = {a,b}. Relation R = {(1,a),(2,a),(3,b)}. Is R a function? Domain & Range?
Solution: Domain = {1,2,3}. Each element of domain maps to exactly one element in B, so R is a function. Range = {a,b}.
Example 2: f: ℤ → ℤ defined by f(x)=x². Is f injective or surjective?
Solution: Not injective (f(2)=f(−2)). Not surjective (no negative integer is a square). So f is many-to-one and not onto.
Add more examples: composition examples, inverse, mapping diagrams, equivalence classes, partitions.
5. Summary & Practice
5.1 Summary Table
| Topic | Key Points |
|---|---|
| Relation | Subset of A×B; properties: reflexive, symmetric, transitive |
| Function | Each domain element maps to exactly one codomain element; domain, range, codomain |
| Injective / Surjective / Bijective | One-to-one, onto, both (invertible) |
| Composition | (g ∘ f)(x) = g(f(x)); composition preserves injectivity and surjectivity under conditions |
| Counting | Relations: 2^(m·n). Functions: n^m. |
5.2 Practice Questions
- Define relation and function with a mapping diagram example.
- Prove: composition of two injective functions is injective.
- Let f: {1,2,3} → {a,b,c}. How many relations and how many functions exist?
- Give an example of an equivalence relation on set {1,2,3} and list equivalence classes.
- If f(x)=2x+1 and g(x)=x², compute (g ∘ f)(x) and (f ∘ g)(x).
FAQ
A: No — a function requires exactly one output per input; relations may allow multiple outputs for a single input.
A: If it is both injective (no two inputs share the same output) and surjective (every codomain value is reached), then it’s bijective.
A: They partition the original set into disjoint equivalence classes; each class collects elements equivalent under the relation.