Definitions [6]
Given two non-empty sets P and Q. The cartesian product P × Q is the set of all ordered pairs of elements from P and Q, i.e., P × Q = { (p,q) : p ∈ P, q ∈ Q } If either P or Q is the null set, then P × Q will also be empty set, i.e., P × Q = Ø
A relation R from a non-empty set A to a non-empty set B is a subset of the cartesian product A × B. The subset is derived by describing a relationship between the first element and the second element of the ordered pairs in A × B. The second element is called the image of the first element.
The set of all first elements of the ordered pairs in a relation R from a set A to a set B is called the domain of the relation R.
The set of all second elements in a relation R from a set A to a set B is called the range of the relation R. The whole set B is called the codomain of the relation R. Note that range ⊂ codomain.
f: X → Y is a function if each element of X is associated with a unique element of Y
- Domain (X): Set of all input values
- Codomain (Y): Set of all possible outputs
- Range: Set of actual output values of f
- Range ⊆ Codomain
Let f: A → B and g: B → C be any two functions. Then, the composition of f and g, denoted by gof, is defined as a function gof: A → C given by
gof(x) = g[f(x)], ∀ x ∈ A
- Domain (gof) = Domain (f)
- g∘f(x) = g(f(x)) → first apply f, then g
Key Points
| Type of Function | Condition | Key Idea |
|---|---|---|
| One-One (Injective) | f(x₁) = f(x₂) ⇒ x₁ = x₂ | Different inputs → different outputs |
| Onto (Surjective) | Range = Codomain | Every element of the codomain is mapped |
| Into Function | Range ⊂ Codomain | Some elements of the codomain are not mapped |
| Many-One Function | x₁ ≠ x₂ but f(x₁) = f(x₂) | Different inputs → same output |
| Bijective Function | One-one + Onto | Perfect mapping (1-1 and onto) |
