Relations and Functions Quiz

Relations and Functions

Relations

• A relation from a set A to a set B is a subset of A × B.
• It can be represented as:
  • R = {(a, b) | a ∈ A, b ∈ B, and a is related to b}
  • Example: Let A = {1, 2, 3} and B = {4, 5}. Then, {(1, 4), (2, 5), (3, 4)} is a relation from A to B.
• Domain of a relation: The set of all first elements of the ordered pairs in the relation.
• Range of a relation: The set of all second elements of the ordered pairs in the relation.

Functions

• A function is a relation from a set A to a set B such that:
  • Every element of A is related to exactly one element of B.
  • A function is denoted as f: A → B, where f is the function and A and B are the domain and codomain respectively.
• Domain of a function: The set of all elements in A.
• Range of a function: The set of all elements in B that are related to some element in A.
• Codomain of a function: The set of all elements in B.

Types of Functions

• One-One Function (Injective): A function is one-one if different elements of the domain have different images in the codomain.
• Onto Function (Surjective): A function is onto if every element of the codomain is the image of some element in the domain.
• One-One Correspondence (Bijective): A function is bijective if it is both one-one and onto.

Inverse of a Function

• If f: A → B is a bijective function, then there exists a function g: B → A such that:
  • f(g(b)) = b for all b ∈ B
  • g(f(a)) = a for all a ∈ A
• The function g is called the inverse of f, denoted as f^(-1).

Composite Functions

• If f: A → B and g: B → C are two functions, then the composite function gof: A → C is defined as:
  • (gof)(a) = g(f(a)) for all a ∈ A