Ordered Pairs

Ordered Pairs:  An ordered pair consists of two objects or elements in a given fixed order.

For example, if A and B are any two sets, then by an ordered pair of elements we mean a pair (a,b) in that order, where a \in A , b \in B .

In the ordered pair (a,b) , a is called the first component (coordinate) and b 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 (a_1,b_1) and (a_2,b_2) are equal iff a_1=a_2 and b_1=b_2 .

i.e (a_1, b_1) = (a_2, b_2) \Leftrightarrow a_1 = a_2 and b_1 = b_2 .

It is evident from this definition that (1,2) \neq (2,1) and (1, 1) \neq (2,2) .

Cartesian Product Of Sets

Cartesian Product Of Sets:  Let A and B be any two non-empty sets. The set of all ordered pairs (a, b) such that a \in A and b \in B is called the Cartesian product of the sets A and B and is denoted by A \times B .

Thus, A \times B = \{ (a,b) : a \in A \ and \  b \in B \} . If A=\phi or B=\phi , then we define A \times B=\phi

Cartesian Product Of Three Sets: Let A, B and C be three sets.Then, A \times B \times C is the set of all ordered triplets having first element from A , second element from B and third element from C .

i.e. A \times B \times C = \{ (a,b,c) : a \in A, b \in B, c \in C \}

Note: It should be noted that A \times B \times C = (A \times B) \times C = A \times (B \times C)

Number Of Elements In The Cartesian Product Of Two Sets

If A and B are two finite sets, then n(A \times B) = n(A) \times n(B)

Note:  (i) If either A or B is an infinite set, then A \times B is an infinite set.  (ii) If A,B,C are finite sets, then n(A \times B \times C) = n(A) \times n(B) \times n(C)

Graphical Representation Of Cartesian Product Of Sets

2019-08-18_18-48-28Let A and B be any two non-empty sets. To represent A \times B graphically, we draw two mutually perpendicular lines, one horizontal and other vertical. On the horizontal line, we represent the elements of set A and on the vertical line, the elements of B . If a \in A , b \in B , we draw a vertical line through a and a  horizontal line through b .
These two lines will meet in a point which will denote the ordered pair (a, b) . In this manner we mark points corresponding to each ordered pair in A \times B .

For Example,  If A = \{ 1, 2, 3 \} and B = \{ 2, 4 \} , then, A \times B = \{ (1, 2), (1, 4), (2,2), (2, 4), (3,2),(3, 4) \} .

Diagrammatic Representation Of Cartesian Product Of Two Sets

2019-08-18_18-59-10In order to represent A \times B by an arrow diagram, we first draw Venn diagrams representing sets A and B one opposite to the other as shown in adjoining figure. Now, we draw line segments starting from each element of A and terminating to each element of set B . If A = \{ 1, 3, 5 \} and B = \{ a, b \} , then the arrow diagram of A \times B is shown in the figure.

Relations

Relation: Let A and B be two sets. Then a relation R from A to B is a subset of A \times B . Thus, R is a relation from A to B \Leftrightarrow R \subseteq A \times B . If R is a relation from a non-void set A to a non-void set B and if (a,b) \in R , then we write a \ R \ b which is read as , “a is related to b by the relation R ” . If (a, b) \notin R , then we write a \  R \ b and we say that a is not related to b by the relation R .

Total Number of Relations: Let A and B be two non-empty finite sets consisting of m and n elements respectively. Then A \times B consists of mn ordered pairs. So, total number of subsets of A \times B is 2^{mn} . Since each subset of A \times B defines a relation from A to B , so total number of relations from A to B is 2^{mn} . Among these 2^{mn}  relations the void relation \phi and the universal relation A \times B are trivial relations from A to B .

Representation of a Relation

A relation from a set A to a set B 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 R .

For example, if  R is a relation from set A = \{ -2,-1, \ldots ,0,1,2 \} to set B=\{ 0, 1, 4, 9, 10 \} by the rule a R b \Leftrightarrow a^2 =b . Then, 0 \ R \ 0, -2 \ R \ 4, -1 \ R \ 1, 1 \ R \ 1 and 2 \ R \ 4 .

So, R can be described in Roster form as follows: R = \{ (0, 0), (- 1, 1), (- 2, 4), (1, 1), (2, 4) \}

Set Builder Form:  In this form the relation R from set A to set B is represented as
R = \{ (a, b): a \in A, b \in B \ and \ a,b \ satisfy \ the \ rule \ which \ associates \ a \ and \ b  \}

For example, if A=\{ 1, 2, 3, 4, 5 \} and B = \{ 1, \frac{1}{2} , \frac{1}{3} , \frac{1}{4} , \frac{1}{5} , \frac{1}{6} , \ldots \} and R is a relation from A to B given by R = \{ (1, 1), (2, \frac{1}{2} ), (3, \frac{1}{3} ), (4, \frac{1}{4} ), (5, \frac{1}{5} ) \}

Then R in set-builder form can be described as follows: R = \{ (a, b) : a \in A , b \in B \ and \ b = \frac{1}{a} \}

2019-08-18_18-59-10By Arrow Diagram: In order to represent a relation from set A to a set B by an arrow diagram, we draw arrows from first components to the second components of all ordered pairs belonging to R .

For example, relation R =\{ 1,a), (1,b), (3,a), (3, b), (5,a), (5, b) \} from set A = \{ 1,3,5 \} to set B = \{ a, b \} can be represented by the arrow diagram shown in figure.

2019-08-18_18-48-28By Lattice:  In this form, the relation R from set A to set B is represented by darkening the dots in the lattice for A \times B which represent the ordered pairs in R .

For example, if R = \{ (1, 2),(2,2),(3,2),(1,4), (2,4),(3,4) \} is a relation from set A = \{ 1,2,3 \} to set B = \{ 2, 4 \} , then R can be represented by the lattice shown in the figure.

Domain and Range of a Relation

Let R be a relation from a set A to a set B . Then the set of all first components or coordinates of the ordered pairs belonging to R is called the domain of R , while the set of all second components or coordinates of the ordered pairs in R is called the range of R .

Thus, Domain (R) = \{ a: (a,b) \in R \} and, Range (R) = \{ b : (a,b) \in R \}

It is evident from the definition that the domain of a relation from A to B is a subset of A and its range is a subset of B . The set B is called the co-domain of relation R .

If A = \{ 1, 3,5,7 \}, B = \{ 2, 4, 6, 8, 10 \} and let R = \{ (1,8), (3, 6), (5, 2),(1, 4) \} be a relation from A to B  Then, Dom (R) = \{ 1, 3,5 \} and Range (R) = \{ 8,6,2, 4 \} .

Relation on a Set: Let A be a non-void  set. Then, a relation from A to itself i.e. a subset of A \times A , is called a relation on set A .

Inverse of a Relations: Let A and B be two sets and let R be a relation from a set A to a set B . Then the inverse of R , denoted by R^{-1} , is a relation from B to A and is defined by R = \{ (b, a): (a, b) \in R \}

Clearly, (a,b) \in R \Rightarrow (b, a) \in R^{-1} . Also, Dom (R) = Range (R^{-1}) and Range (R) = Dom (R^{-1}) .

For example, Let A = \{ 1, 2, 3 \} B = \{ a, b, c, d \} be two sets and let R = \{ (1, a), (1,c), (2, d), (2, c) \} be a relation from A to B . Then R^{-1} = \{ (a, 1), (c, 1), (d, 2), (c, 2) \} is a relation from B to A

Also, Dom (R) = \{1,2 \} = Range (R^{-1}) , and Range (R) = {a, c, d} = Domain (R^{-1}) .