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Question
Determine whether the following relation is reflexive, symmetric and transitive:
Relation R in the set A = {1, 2, 3, 4, 5, 6} as R = {(x, y) : y is divisible by x}.
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Solution
A = {1, 2, 3, 4, 5, 6}
R = {(x, y) : y is divisible by x}
We know that any number (x) is divisible by itself.
⇒ (x, x) ∈ R
∴ R is reflexive.
Now, (2, 4) ∈ R ....[As 4 is divisible by 2.]
But (4, 2) ∉ R ....[As 2 is not divisible by 4.]
∴ R is not symmetric.
Let (x, y), (y, z) ∈ R. Then, y is divisible by x and z is divisible by y.
∴ z is divisible by x.
⇒ (x, z) ∈ R
∴ R is transitive.
Hence, R is reflexive and transitive but not symmetric.
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