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Question
Determine whether the following relation is reflexive, symmetric and transitive:
Relation R in the set N of natural numbers defined as R = {(x, y) : y = x + 5 and x < 4}.
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Solution
R = {(x, y) : y = x + 5 and x < 4} = {(1, 6), (2, 7), (3, 8)}
It is seen that (1, 1) ∉ R.
∴ R is not reflexive.
(1, 6) ∈ R but (1, 6) ∉ R.
∴ R is not symmetric.
Now, since there is no pair in R such that (x, y) and (y, z) ∈ R, we need to look for the ordered pair (x, y).
∴ R is transitive.
Hence, R is neither reflexive, nor symmetric, but it is transitive.
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