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Question
An integer m is said to be related to another integer n if m is a multiple of n. Check if the relation is symmetric, reflexive and transitive.
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Solution
R={ (m, n) : m, n ∈ Z, m=kn, where k ∈ N}
Reflexivity :
Let m be an arbitrary elementof R.Then,
m = km is true for k=1
⇒ (m, m) ∈ R
Thus, R is reflexive.
Symmetry: Let (m, n) ∈ R
⇒ m =kn for some k ∈ N
→ `n =1 /km`
⇒ (n, m) ∉ R
Thus, R is not symmetric.
Transitivity : Let (m, n) and (n, o) ∈ R
⇒ m=kn and n=lo for some k, l ∈ N
⇒ m=(kl) o
Here, kl ∈ R
⇒ (m, o) ∈ R
Thus, R is transitive.
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