Chapter 2 Relations and Functions
Ordered Pair
An ordered pair consists of two objects or elements in a given fixed order.
Equality of Two Ordered Pairs
Two ordered pairs (a, b) and (c, d) are equal if a = c and b = d.
Cartesian Product of Two Sets
For any two non-empty sets A and B, the set of all ordered pairs (a, b) where a ∈ A and b ∈ B is called the
cartesian product of sets A and B and is denoted by A × B.
Thus, A × B = {(a, b) : a ∈ A and b ∈ B}
If A = Φ or B = Φ, then we define A × B = Φ
Note:
A×B≠B×A
If n(A) = m and n(B) = n, then n(A × B) = mn and n(B × A) = mn
If atieast one of A and B is infinite, then (A × B) is infinite and (B × A) is infinite.
Relations
A relation R from a non-empty set A to a non-empty set B is a subset of the cartesian product set 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 set of all first elements in a relation R is called the domain of the relation B, and the set of all second
elements called images is called the range of R.
Note:
A relation may be represented either by the Roster form or by the set of builder form, or by an arrow diagram
which is a visual representation of relation.
If n(A) = m, n(B) = n, then n(A × B) = mn and the total number of possible relations from set A to set B = 2mn
Inverse of Relation
For any two non-empty sets A and B. Let R be a relation from a set A to a set B. Then, the inverse of relation R,
denoted by R-1 is a relation from B to A and it is defined by
R-1 ={(b, a) : (a, b) ∈ R}
Domain of R = Range of R-1 and
Range of R = Domain of R-1.
Functions
A relation f from a set A to set B is said to be function, if every element of set A has one and only image in set
B.
In other words, a function f is a relation such that no two pairs in the relation have the first element.
A function f : A → B is called a real-valued function if B is a subset of R (set of all real numbers). If A and B both
are subsets of R, then f is called a real function.
Ordered Pair
An ordered pair consists of two objects or elements in a given fixed order.
Equality of Two Ordered Pairs
Two ordered pairs (a, b) and (c, d) are equal if a = c and b = d.
Cartesian Product of Two Sets
For any two non-empty sets A and B, the set of all ordered pairs (a, b) where a ∈ A and b ∈ B is called the
cartesian product of sets A and B and is denoted by A × B.
Thus, A × B = {(a, b) : a ∈ A and b ∈ B}
If A = Φ or B = Φ, then we define A × B = Φ
Note:
A×B≠B×A
If n(A) = m and n(B) = n, then n(A × B) = mn and n(B × A) = mn
If atieast one of A and B is infinite, then (A × B) is infinite and (B × A) is infinite.
Relations
A relation R from a non-empty set A to a non-empty set B is a subset of the cartesian product set 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 set of all first elements in a relation R is called the domain of the relation B, and the set of all second
elements called images is called the range of R.
Note:
A relation may be represented either by the Roster form or by the set of builder form, or by an arrow diagram
which is a visual representation of relation.
If n(A) = m, n(B) = n, then n(A × B) = mn and the total number of possible relations from set A to set B = 2mn
Inverse of Relation
For any two non-empty sets A and B. Let R be a relation from a set A to a set B. Then, the inverse of relation R,
denoted by R-1 is a relation from B to A and it is defined by
R-1 ={(b, a) : (a, b) ∈ R}
Domain of R = Range of R-1 and
Range of R = Domain of R-1.
Functions
A relation f from a set A to set B is said to be function, if every element of set A has one and only image in set
B.
In other words, a function f is a relation such that no two pairs in the relation have the first element.
A function f : A → B is called a real-valued function if B is a subset of R (set of all real numbers). If A and B both
are subsets of R, then f is called a real function.
