Ordered Pairs: An ordered pair consists of two objects or elements in a given fixed order.
For example, if and are any two sets, then by an ordered pair of elements we mean a pair in that order, where , .
In the ordered pair , is called the first component (coordinate) and is called the second component (coordinate).
Note: An ordered pair is not a set consisting of two elements. The ordering of the two elements in an ordered pair is important and the two elements need not be distinct.
Equality Of Ordered Pairs: Two ordered pairs and are equal iff and .
i.e and .
It is evident from this definition that and .
Cartesian Product Of Sets
Cartesian Product Of Sets: Let and be any two non-empty sets. The set of all ordered pairs such that and is called the Cartesian product of the sets and and is denoted by .
Thus, . If or , then we define
Cartesian Product Of Three Sets: Let and be three sets.Then, is the set of all ordered triplets having first element from , second element from and third element from .
Note: It should be noted that
Number Of Elements In The Cartesian Product Of Two Sets
If and are two finite sets, then
Note: (i) If either or is an infinite set, then is an infinite set. (ii) If are finite sets, then
Graphical Representation Of Cartesian Product Of Sets
Let and be any two non-empty sets. To represent graphically, we draw two mutually perpendicular lines, one horizontal and other vertical. On the horizontal line, we represent the elements of set and on the vertical line, the elements of . If , , we draw a vertical line through and a horizontal line through .
These two lines will meet in a point which will denote the ordered pair . In this manner we mark points corresponding to each ordered pair in .
For Example, If and , then, .
Diagrammatic Representation Of Cartesian Product Of Two Sets
In order to represent by an arrow diagram, we first draw Venn diagrams representing sets and one opposite to the other as shown in adjoining figure. Now, we draw line segments starting from each element of and terminating to each element of set . If and , then the arrow diagram of is shown in the figure.
Relation: Let and be two sets. Then a relation from to is a subset of . Thus, is a relation from to . If is a relation from a non-void set to a non-void set and if , then we write which is read as , “ is related to by the relation ” . If , then we write
R and we say that a is not related to by the relation .
Total Number of Relations: Let and be two non-empty finite sets consisting of and elements respectively. Then consists of ordered pairs. So, total number of subsets of is . Since each subset of defines a relation from to , so total number of relations from to is . Among these relations the void relation and the universal relation are trivial relations from to .
Representation of a Relation
A relation from a set to a set can be represented in, any one of the following forms:
Roster Form: In this form a relation is represented by the set of all ordered pairs belonging to .
For example, if is a relation from set to set by the rule . Then, and .
So, can be described in Roster form as follows:
Set Builder Form: In this form the relation from set to set is represented as
For example, if and and is a relation from to given by
Then in set-builder form can be described as follows:
By Arrow Diagram: In order to represent a relation from set to a set by an arrow diagram, we draw arrows from first components to the second components of all ordered pairs belonging to .
For example, relation from set to set can be represented by the arrow diagram shown in figure.
By Lattice: In this form, the relation from set to set is represented by darkening the dots in the lattice for which represent the ordered pairs in .
For example, if is a relation from set to set , then can be represented by the lattice shown in the figure.
Domain and Range of a Relation
Let be a relation from a set to a set . Then the set of all first components or coordinates of the ordered pairs belonging to is called the domain of , while the set of all second components or coordinates of the ordered pairs in is called the range of .
Thus, Domain and, Range
It is evident from the definition that the domain of a relation from to is a subset of and its range is a subset of . The set is called the co-domain of relation .
If and let be a relation from to Then, Dom and Range .
Relation on a Set: Let be a non-void set. Then, a relation from to itself i.e. a subset of , is called a relation on set .
Inverse of a Relations: Let and be two sets and let be a relation from a set to a set . Then the inverse of , denoted by , is a relation from to and is defined by
Clearly, . Also, Dom Range and Range .
For example, Let be two sets and let be a relation from to . Then is a relation from to
Also, Dom Range , and Range Domain .