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GATE | GATE-IT-2004 | Question 4
  • Last Updated : 14 Feb, 2018

Let R1 be a relation from A = {1, 3, 5, 7} to B = {2, 4, 6, 8} and R2 be another relation from B to C = {1, 2, 3, 4} as defined below:

  1. An element x in A is related to an element y in B (under R1) if x + y is divisible by 3.
  2. An element x in B is related to an element y in C (under R2) if x + y is even but not divisible by 3.

Which is the composite relation R1R2 from A to C?

 
(A) R1R2 = {(1, 2), (1, 4), (3, 3), (5, 4), (7, 3)}
(B) R1R2 = {(1, 2), (1, 3), (3, 2), (5, 2), (7, 3)}
(C) R1R2 = {(1, 2), (3, 2), (3, 4), (5, 4), (7, 2)}
(D) R1R2 = {(3, 2), (3, 4), (5, 1), (5, 3), (7, 1)}


Answer: (C)

Explanation:
R1 is a relation from A = {1, 3, 5, 7} to B = {2, 4, 6, 8} .
Under R1, an element x in A is related to an element y in B if x + y is divisible by 3.

Thus, R1 = {(1, 2), (1, 8), (3, 6), (5, 4), (7, 2), (7, 8)}

R2 is a relation from B = {2, 4, 6, 8} to C = {1, 2, 3, 4}
Under R2, an element y in B is related to an element z in C if y + z is even but not divisible by 3.

Thus, R2 = {(2, 2), (4, 4), (6, 2), (6, 4), (8, 2)}

Then the composition of R1 with R2, denoted R2R1, is the relation from A to C defined by the following property: (x, z) \epsilon R2R1 If and only if there is a y E B such that (x, y) \epsilon R1 and (y, z) \epsilon R2.

Thus, R1R2 = {(1, 2), (3, 2), (3, 4), (5, 4), (7, 2)}

 
Thus, option (C) is correct.

 
Please comment below if you find anything wrong in the above post.


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