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Question
Determine whether the following relation is reflexive, symmetric and transitive:
Relation R in the set Z of all integers defined as R = {(x, y) : x − y is an integer}.
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Solution
R = {(x, y) : x − y is an integer}
Now, for every x ∈ Z, (x, x) ∈ R as x − x = 0 is an integer.
∴ R is reflexive.
Now, for every x, y ∈ Z, if (x, y) ∈ R, then x − y is an integer.
⇒ −(x − y) is also an integer.
⇒ (y − x) is an integer.
∴ (y, x) ∈ R
∴ R is symmetric.
Now, let (x, y) and (y, z) ∈ R, where x, y, z ∈ Z.
⇒ (x − y) and (y − z) are integers.
⇒ x − z = (x − y) + (y − z) is an integer.
∴ (x, z) ∈ R
∴ R is transitive.
Hence, R is reflexive, symmetric and transitive.
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