Relations and Functions in Math

Relations and Functions

Relations

• A relation R from set A to set B is a subset of A × B
• It is denoted by R(A, B) or simply R
• If (a, b) ∈ R, then a is related to b by R
• Domain of R: set of all elements in A that are related to some element in B
• Range of R: set of all elements in B that are related to some element in A

Functions

• A function f from set A to set B is a relation that satisfies:
  • Every element in A is related to exactly one element in B
  • Domain of f = A
• It is denoted by f: A → B or f(x)
• f(a) is the image of a under f
• Range of f: set of all images of elements in A

Types of Functions

• One-One Function (Injective): a function f: A → B is one-one if every element in B is the image of at most one element in A
• Onto Function (Surjective): a function f: A → B is onto if every element in B is the image of at least one element in A
• One-One Onto Function (Bijective): a function f: A → B is one-one onto if it is both one-one and onto
• Constant Function: a function f: A → B is constant if every element in A has the same image in B
• Identity Function: a function f: A → A is identity if every element in A is mapped to itself

Relations and Functions

Relations

• A relation R is a subset of the Cartesian product of set A and set B, denoted by R(A, B) or simply R.
• If (a, b) is an element of R, it means a is related to b by R.
• The domain of R is the set of all elements in A that are related to some element in B.
• The range of R is the set of all elements in B that are related to some element in A.

Functions

• A function f is a relation that satisfies two conditions: every element in A is related to exactly one element in B, and the domain of f is equal to A.
• It is denoted by f: A → B or f(x), where f(a) is the image of a under f.
• The range of f is the set of all images of elements in A.

Types of Functions

One-One Function (Injective)

• A function f: A → B is one-one if every element in B is the image of at most one element in A.
• In other words, every element in B has a unique pre-image in A.

Onto Function (Surjective)

• A function f: A → B is onto if every element in B is the image of at least one element in A.
• In other words, every element in B has a pre-image in A.

One-One Onto Function (Bijective)

• A function f: A → B is one-one onto if it is both one-one and onto.
• In other words, every element in B has a unique pre-image in A, and every element in A has a unique image in B.

Constant Function

• A function f: A → B is constant if every element in A has the same image in B.
• In other words, f(a) = f(b) for all a, b in A.

Identity Function

• A function f: A → A is identity if every element in A is mapped to itself.
• In other words, f(a) = a for all a in A.