Let R1 be a relation from A = {1, 3, 5, 7} to B = {2, 4, 6, 8} and R2 be…

2004

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?

  1. A.

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

  2. B.

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

  3. C.

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

  4. D.

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

Attempted by 118 students.

Show answer & explanation

Correct answer: C

Solution summary: The composite relation is {(1, 2), (3, 2), (3, 4), (5, 4), (7, 2)}.

  • Step 1: Find R1 from A to B (x+y divisible by 3).

    Compute pairs: (1,2), (1,8), (3,6), (5,4), (7,2), (7,8).

  • Step 2: Find R2 from B to C (b+c even but not divisible by 3).

    Since all b in B are even, c must be even (2 or 4). Check divisibility by 3:

    Valid pairs: (2,2), (4,4), (6,2), (6,4), (8,2).

  • Step 3: Form the composite R1R2. Include (a,c) when there exists b with (a,b) in R1 and (b,c) in R2.

    Use the lists from Steps 1 and 2:

    1: via b = 2 and 8, and both 2 and 8 relate to 2 in R2, so (1,2).

    3: via b = 6, and 6 relates to 2 and 4, so (3,2) and (3,4).

    5: via b = 4, and 4 relates to 4, so (5,4).

    7: via b = 2 and 8, and both relate to 2, so (7,2).

  • Final answer: R1R2 = {(1, 2), (3, 2), (3, 4), (5, 4), (7, 2)}.

Explore the full course: Gate Guidance By Sanchit Sir