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Question
Given an example of a relation. Which is
(ii) Transitive but neither reflexive nor symmetric.
Given an example of a relation. Which is
(ii) Transitive but neither reflexive nor symmetric.
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Solution
R = {(a, b): a < b}
For any a∈R, we have (a,a)/∈R since a cannot be strictly less than a (itself). Therefore, R is not reflexive. [∵a=a]
Now, (1,2)∈R (as 1 < 2)
But, (2,1)/∈R as 2 not less than 1
Now, let (a, b), (b, c) ∈R.
⇒a<b and b<c⇒a<c⇒(a,c)∈R.
Therefore, R is transitive.
Hence, relation R is transitive but neither symmetric nor reflexive.
R = {(a, b): a < b}
For any a∈R, we have (a,a)/∈R since a cannot be strictly less than a (itself). Therefore, R is not reflexive. [∵a=a]
Now, (1,2)∈R (as 1 < 2)
But, (2,1)/∈R as 2 not less than 1
Now, let (a, b), (b, c) ∈R.
⇒a<b and b<c⇒a<c⇒(a,c)∈R.
Therefore, R is transitive.
Hence, relation R is transitive but neither symmetric nor reflexive.
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