Ordered Pair

Two elements a \ and \ b  listed in a specific order, form an ordered pair, denoted by a \ and \ b .

In an ordered pair  a \ and \ b , we call a as the first component and b as the second component.

By changing the positions of the components, the ordered pair is changed, i.e. (5, \ 4) \ne (4, \ 5)

Also if (a, \ b) = (c, d) \Rightarrow a=c \ and \ b=d  

Thus (a, \ b) = (2, 7) \Rightarrow a=2 \ and \ b=7  

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Cartesian Product of Two Setsm34x

Let  A \ and \ B  be two non-empty sets. Then their Cartesian product A \ \times \ B   is the set of all ordered pairs  a \ and \ b  such that a \in A \ and\  b\in B.

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Example 1:

Let A=\{a,\ b \} \ and \ B=\{1, \ 2, \ 3\}  

Then A \times B = \{a,\ b \} \times \{1, \ 2, \ 3\} = \{ (a, 1), \ (a, 2),\  (a, 3),\  (b, 1),\  (b, 2),\  (b, 3)\}  

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Relation

Let A \ and \ B   be two non-empty sets. Then every subset of  A \times B  is called a relation from A \ to\  B  . i.e., if  R \subseteq A\times B , then  R   is a relation from A \ to \ B 

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Representation of a Relation

Roster Form:  When a relation is represented by the set of all ordered pairs contained in it, then it is said to be in roster form.

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 Example 2:

If A=\{1, 9, 16, 25\} \ and\ B=\{1, 2, 3, 4, 5\}    then a relation R \ from\ A \ to\  B   defined as ‘is the square of’ can be represented in the roster form as:

R = \{(1,1), (9,3), (16, 4), (25, 5)\}

 You will notice that the first component of each subset is square of the second component of the subset.

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Arrow Diagram: Let  R   be a relation from A \ to \ B . We can draw the sets as shown pictorially.  Then, we draw arrows from A \ to \ B  to indicate the pairing of the corresponding elements related to each other. Thus, we can show the relation given in Example 2 by the arrow diagram as shown.

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Set Builder Form: A relation  R   from A \ to \ B  is said to be in a Set-Builder form when written as:

R=\{(a,\ b): a \in A,\ b \in B\ and\ a \ is \ connected \ with \ b \ by \ a \ given rule\}

The relation given in Example 2 can be represented in the set-builder form as:

R=\{(a,\ b): a \in A,\ b \in B\ and\ a=b^2\}  

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Domain and Range of a Relation

Let  R   be a relation from A \ to \ B . Then,

Domain (R ) = Set of first components of all ordered pairs in  R  

Range (R ) = Set of second components of all ordered pairs in  R  

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Rule of Relation

m36xThe property that tells us how the first component is related to the second component of each ordered pair in  R  , is called the rule of the relation.

Let’s look at the following example:

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Example 3:

Let  A=\{2,\  3,\  4\} \ and\ B=\{1, \ 3, \ 5\}  

Let R   be the relation ‘is less than’ from  A \ to \ B . Find R  . Also, write down the domain and range of  R  .

Solution:

We have:

R=\{(a,\ b): a \in A,\ b \in B\ and\ a  < b \} = \{(2,\ 3),\  (2,\ 5),\  (3,\ 5),\  (4,\ 5) \}  

Note: We take only those pairs  (a, \ b) \ of A\times B  in which a < b

Hence, R = \{(2,3), \ (2,5), \ (4,5) \}  

Domain ( R ) = Set of first components of elements of  R = \{2,\  3,\  4\}

Range ( R ) = Set of second components of elements of   R = \{3,\ 5\}

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