Aptitude
October 5, 2023GATE 2004IT
October 5, 2023GATE 2004IT
Question 4

Let R_{1} be a relation from A = {1, 3, 5, 7} to B = {2, 4, 6, 8} and R_{2} 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 R_{1}) if x + y is divisible by 3.
2. An element x in B is related to an element y in C (under R_{2}) if x + y is even but not divisible by 3.
Which is the composite relation R_{1}R_{2} from A to C?
R_{1}R_{2} = {(1, 2), (1, 4), (3, 3), (5, 4), (7, 3)}


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


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


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

Question 4 Explanation:
From the given information,
R_{1} ={(1,2), (1,8), (3,6), (5,4), (7,2), (7,8)}
where x+y is divisible by 3
R_{2} = {(2,2), (4,4), (6,2), (6,4), (8,2)}
where x+y is not divisible by 3
Then the composition of R_{1} with R_{2} denotes R_{1}R_{2}, is the relation from A to C defined by property such as:
(x,z) ∈ R_{1}R_{2}, iff if there is a y ∈ B such that (x,y) ∈ R_{1} and (y,z) ∈ R_{2}.
Thus, R_{1}R_{2} = {(1,2), (3,2), (3,4), (5,4), (7,2)}
R_{1} ={(1,2), (1,8), (3,6), (5,4), (7,2), (7,8)}
where x+y is divisible by 3
R_{2} = {(2,2), (4,4), (6,2), (6,4), (8,2)}
where x+y is not divisible by 3
Then the composition of R_{1} with R_{2} denotes R_{1}R_{2}, is the relation from A to C defined by property such as:
(x,z) ∈ R_{1}R_{2}, iff if there is a y ∈ B such that (x,y) ∈ R_{1} and (y,z) ∈ R_{2}.
Thus, R_{1}R_{2} = {(1,2), (3,2), (3,4), (5,4), (7,2)}
Correct Answer: C
Question 4 Explanation:
From the given information,
R_{1} ={(1,2), (1,8), (3,6), (5,4), (7,2), (7,8)}
where x+y is divisible by 3
R_{2} = {(2,2), (4,4), (6,2), (6,4), (8,2)}
where x+y is not divisible by 3
Then the composition of R_{1} with R_{2} denotes R_{1}R_{2}, is the relation from A to C defined by property such as:
(x,z) ∈ R_{1}R_{2}, iff if there is a y ∈ B such that (x,y) ∈ R_{1} and (y,z) ∈ R_{2}.
Thus, R_{1}R_{2} = {(1,2), (3,2), (3,4), (5,4), (7,2)}
R_{1} ={(1,2), (1,8), (3,6), (5,4), (7,2), (7,8)}
where x+y is divisible by 3
R_{2} = {(2,2), (4,4), (6,2), (6,4), (8,2)}
where x+y is not divisible by 3
Then the composition of R_{1} with R_{2} denotes R_{1}R_{2}, is the relation from A to C defined by property such as:
(x,z) ∈ R_{1}R_{2}, iff if there is a y ∈ B such that (x,y) ∈ R_{1} and (y,z) ∈ R_{2}.
Thus, R_{1}R_{2} = {(1,2), (3,2), (3,4), (5,4), (7,2)}
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