Verify whether the relation "a divides b" written as alb is reflexive, symmetric or transitive on N. Also state if its equivalence relation or not
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Verify whether the relation "a divides b" written as alb is reflexive, symmetric or transitive on N. Also state if its equivalence relation or not
Answer :
R is reflexive, transitive but not symmetric.
R is not equivalence.
Concept to be used :
Solution :
Here,
The given relation is R = {(a,b) : a|b where a, b ∈ N}
Whether R is reflexive or not ?
We know that, every natural number divides itself.
Thus,
→ a|a ∀ a ∈ N
→ (a,a) ∈ R ∀ a ∈ N
→ R is reflexive.
Whether R is symmetric or not ?
We have 2 ∈ N and 4 ∈ N.
We know that, 2|4 but 4∤2.
→ (2,4) ∈ R but (4,2) ∉ R.
→ R is not symmetric.
Whether R is transitive or not ?
Let (a,b) ∈ R and (b,c) ∈ R, where a, b, c ∈ N.
→ a|b and b|c
→ b = ma and c = nb, where m, n ∈ N.
→ c = n × ma, where m, n ∈ N.
→ c = mna, where m, n ∈ N.
→ a|c
→ (a,c) ∈ R
i.e.
(a,b) ∈ R, (b,c) ∈ R → (a,c) ∈ R ∀ a, b, c ∈ N.
→ R is transitive.
Since R is reflexive, transitive but not symmetric, hence R is not equivalence.