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How to proceed in such questions.

Consider set $R = {(1, 1), (2, 1), (3, 2), (4, 3)}$. If set $P = R^1 \cup R^2 \cup R^3,  $ then which of the following represents the set P? $(R^2 = R^1oR^1 i.e.$ composition of R with R)

A {(1, 1) (2, 1) (3, 2) (4, 1) (4,2), (4,3), (3,1)

B  {(1, 1) (2, 1) (3, 2) (4, 1) (4,2), (4,3), (3,3)

C  {(1, 1) (2, 1) (3, 2) (4, 4) (4,2), (4,3), (3,1)

D {(1, 1) (2, 1) (2, 2) (4, 1) (4,2), (4,3), (3,1)
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2 Answers

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$R = (1,1) , (2,1) ,(3,2),(4,3) = R^{1}$

$R^{2} = R^{1}\circ R^{1} = (1,1) , (2,1) , (3,1) , (4,2)$

$R^{3} = R^{1}\circ R^{1} \circ R^{1} = (1,1) , (2,1) , (3,1) , (4,1)$

ans A

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the concept behind this is

suppose I take an small example  if X =  {(1,1) , (1,2), (2,6),}   and Y  = {(1,6),(2,1), (4,0)}

what is Y $\circ$ X ==== > X composed with  Y

X relates 1 ---> 1

Y relates 1 ---->6

then , the composition should relate (1,6)

again ,

X relates 1 ---->2

Y relates 2----->1

then , the composition should relalate (1,1)

again ,

X relates 2 ----->6

Y doesn't relates related to anything

 

therefore , composition formed (1,6) , (1,1)
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2 Comments

@Magma How you got (2, 1) in second line in the answer. I am not getting this subset?
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R relates 2 ---- >1

R relates 2 -----> 1

(2,1)
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Ans. is A

For composition of sets containing pair of elements:

Take first pair from first set say (x1,y1), then pick second element y1 of this pair and check it in the second set of R say (x2,y2). If there is any pair starting with x2=y1, if yes then take its second element y2 and make a subset of (x1,y2) in RoR similarly check for all other pairs from first set.

 

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