Civil Engineering (CE) Exam > Civil Engineering (CE) Questions > Let m ∈ Z and consider the relation Rm d...
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Let m ∈ Z and consider the relation Rm defined by a Rm b if and only if a ≡ b mod m. Then Rm is -
- a)reflexive relation
- b)symmetric relation
- c)transitive relation
- d)equivalence relation
Correct answer is option 'D'. Can you explain this answer?
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Let m ∈ Z and consider the relation Rm defined by a Rm b if and o...
Be a positive integer. If m is even, then m can be written in the form m = 2k for some positive integer k. If m is odd, then m can be written in the form m = 2k + 1 for some positive integer k.
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Let m ∈ Z and consider the relation Rm defined by a Rm b if and o...
Given: a Rmb if and only if a ≡ b mod.
Since m|(a-a) = 0, we have a ≡ a mod m
So Rm is reflexive.
if m|(a-b), then
⇒ m|(-1)(a-b).
⇒ m|(b-a).
⇒ So, if a ≡ b mod m ⇒ b ≡ a mod m.
Since m|(a-a) = 0, we have a ≡ a mod m
So Rm is reflexive.
if m|(a-b), then
⇒ m|(-1)(a-b).
⇒ m|(b-a).
⇒ So, if a ≡ b mod m ⇒ b ≡ a mod m.
- Therefore, Rm is symmetric.
- If aRmb and bRmc
⇒ m|(a-b) and m|(b - c).
⇒ m|[(a-b) + (b-c)]
⇒ m|(a-c) ⇒ a Rmc
⇒ m|[(a-b) + (b-c)]
⇒ m|(a-c) ⇒ a Rmc
Therefore Rm is transitive.
Hence the relation is an equivalence relation.
Therefore option 4 is correct.
Hence the relation is an equivalence relation.
Therefore option 4 is correct.
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Let m ∈ Z and consider the relation Rm defined by a Rm b if and only if a ≡ b mod m. Then Rm is -a)reflexive relationb)symmetric relationc)transitive relationd)equivalence relationCorrect answer is option 'D'. Can you explain this answer?
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