How to Discuss the Relation For Reflexive Symmetric or Transitive :
Here we are going to see how to discuss the relation for reflexive symmetric or transitive.
Let S be any non-empty set. Let R be a relation on S. Then
Question 1 :
Discuss the following relations for reflexivity, symmetricity and transitivity:
(i) The relation R defined on the set of all positive integers by “mRn if m divides n”.
Solution :
Condition for reflexive :
R is said to be reflexive, if a is related to a for a ∈ S.
Every element is divided by itself, for all m ∈ R, m divides m and for all n ∈ R n divides n.
Hence it is reflexive.
Condition for symmetric :
R is said to be symmetric if a is related to b implies that b is related to a.
From the given question, we come to know that m divides n, but the vice versa is not true.
For example, if m = 2 and n = 4 ∈ R, then we may say that 2 divides 4. But we cannot say that 4 divides 2. Hence symmetric is not true.
Condition for transitive :
R is said to be transitive if “a is related to b and b is related to c” implies that a is related to c.
If m divides n, then n = mk ----(1)
If n divide p, then p = nq ----(2)
Here "k" and "q" are constants.
By applying the value of "n" in (2), we get
p = m k q
From this, we come to know that p is the multiple of m. So, it is transitive.
Hence the given relation is reflexive, not symmetric and transitive.
Difference between reflexive and identity relation
After having gone through the stuff given above, we hope that the students would have understood, how to check whether a relation is reflexive, symmetric and transitive.
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