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
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