### CHAPTER AT A GLANCE

**Ordered pair**– A pair of objects (elements) taken (listed) in a definite (specific) order is known as an ordered pair. In the ordered pair (ai, bi), ai is known as first element and bi is known as second element.**Equality of ordered pairs**– Two ordered pairs (a1, b1) and (a2, b2) are equal iff a1 = a2 and b1 = b2**Cartesian product**– Let A and B be two non-empty sets, then the set of all distinct ordered pair of the form (ai, bi), where a1 ÎA and b1 ÎB is called

cartesian product of A and B and is denoted by A × B. If A and B have m and

n elements respectively, then, number of order pairs in A × B is equal to m × n.**Relations**– Let A and B be two non-empty sets then every subset of A × B

defines a relation from A to B and every relation from A to B is a subset of A × B.**Domain (D) and Range (R) of a Relation**– If R be a relation from set A to B

then the set of first entries of all ordered pairs in R is called domain and set of

all second entries in all order pairs of R is called range of R.**Inverse Relation**– Let A and B be two sets and let R be a relation from set A to

set B then the inverse of R, denoted by R–1, is a relation from set B to set A

and is defined as.

**Number of Relation**– Let A and B be two non-empty finite sets having m and

n elements respectively then the number of relations from A to B is (2)m×n**Universal Relation**– A Relation R in a set A is called a universal relation if R = A × A**Empty Relation**: A relation R in a set A is called an empty relation, if no element

of A is related to any element of A.

**Equivalence Relation**– Any relation on a set A which is Reflexive, Symmetric and Transitive is called an equivalence Relation.

**Function**: A function from a non-empty set A to a non-empty set B is a relation

from set A to set B such that each element of set A is associated with one and

only one element (i.e. unique element) of set B (i.e. each element of set A is

associated with a unique element of set B)

Note that more than one element of A may be associated with any element of

B but in set B, there may be one or more elements to which no element of set A

is associated.**Domain**– The set A is called the domain of ‘f’ (denoted by Df )**Co-domain**– The set B is called the co-domain of ‘f ’ (denoted by Cf)**Range**– The range of ‘f ’ denoted by Rf is the set consisting of all the image of the elements of the domain A.

**One-One function (Injective)**– Let f : A ® B then f is called a one-one function

if no two different elements in A have the same image i.e. different elements in

An associated with different elements in B.**Onto function (Surjective)**: A function f : A ® B is said to be onto function,

if every element of B is the image of some element of A under f.

Note : f : A ® B is onto iff Range of f = B.**One-One Onto function (Bijective)**: A function f : A –> B is said to be one-

one and onto if f is both one-one and onto.**Many-One – A function f**: A –> B is said to be many-one if two or more elements

of set A have the same image in B.**Even and Odd function**– A function f(x) is said to be even function iff

f(–x) = f(x) and A function f(x) is said to be odd function iff f (–x) = – f(x).**Composition functions**– Let f : A –> B and g : B –> C be two functions. Then the composition of f and g, denoted by gof : A –> C given by

**Invertible Function and Inverse Function:**A function f : X -> Y is defined to be invertible if there exists a function g : Y ® X such that g of = Ix and fog = Iy . The function g is called the inverse of f and is denoted by f–1 . Thus, f is invertible, then f must be one-one and onto and conversely, if f is one-one and onto, then f must be invertible.

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