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CHAPTER 6 : RELATIONS, FUNCTIONS AND VARIATION

6.1 Relations

Ordered Pairs

An ordered pair is a pair of elements, where the order of elements is definite. It is written by enclosing the pair of elements in brackets as ( x, y ). In an ordered pair the two members need not be different. For example ( 7, 7 ).

If ( x1 ; y1 ) and ( x2, y2 ) represent ordered pairs then ( x1, y1) = ( x2, y2 ) if and only if x1 = x2 and y1 = y2. But ( 5, 9 ) ¹
( 9, 5 ).

Cartesian product of two set

Let A and B be two sets. The cartesian product of two sets A and B is the set of all ordered pairs in which the first element of every pair belongs to the set A and the second belongs to set B. It is written as A ´ B and read as A cross B.

i.e. A ´ B =

Example If A = { 1, 2, 3 }, B = { 4, 5, 6 } then find A ´ B and B ´ A

Solution : A ´ B = { ( 1, 4 ), ( 1, 5 ), ( 1, 6 ), ( 2 , 4 ),
( 2, 5 ), ( 2 , 6 ), ( 3, 4 ), ( 3, 5 ), ( 3, 6) }

Also B ´ A = { ( 4, 1 ), ( 4, 2 ), ( 4, 3 ), ( 5, 1 ), ( 5, 2 ),
( 5, 3), ( 6, 1 ), ( 6, 2 ), ( 6, 3 ) }

Clearly A ´ B ¹ B ´ A

Example If A = { x | x Î N Ù x £ 3 } , write A ´ A and graph it.

Solution : A = { x | x Î N Ù x £ 3 } = { 1, 2, 3 }

Therefore A ´ A = { 1, 2, 3 } ´ { 1, 2, 3 }

i.e. A ´ B = { (1, 1), (1, 2 ), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3,3 ) }

Index

6.1 Relations
6.2 Functions
6.3 Variation

Chapter 7

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