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Consider the following relations in 1, then which of the following is an equivalence relation?
- a)m~n if and only if mn ≥ 0
- b)m~n if and only if mn > 0
- c)m~n if and only if |m| = |n|
- d)m~n if and only if mn ≤ 0
Correct answer is option 'C'. Can you explain this answer?
Most Upvoted Answer
Consider the following relations in 1, then which of the following is ...
Relation containing less than or greater than sign is never equivalence relation. so option c is correct.
Reflexivity : |m|=|m| for all m ==> relation is reflexive
Symmetry : if |m|=|n| then |n|=|m| ==> relation is symmetric
Transitivity : if |m|=|n| and |n|=|p| then |p|=|m| ==> relation is transitive
hence |m|=|n| is an equivalence relation
Reflexivity : |m|=|m| for all m ==> relation is reflexive
Symmetry : if |m|=|n| then |n|=|m| ==> relation is symmetric
Transitivity : if |m|=|n| and |n|=|p| then |p|=|m| ==> relation is transitive
hence |m|=|n| is an equivalence relation
Community Answer
Consider the following relations in 1, then which of the following is ...
To be an equivalence relation, a relation must satisfy three properties: reflexive, symmetric, and transitive.
a) m~n if and only if mn is divisible by 5.
To check if it is reflexive, we need to see if for any m, m~m. If we choose m = 5, then 5 * 5 = 25, which is divisible by 5. Therefore, it is reflexive.
To check if it is symmetric, we need to see if for any m and n, if m~n, then n~m. If m~n, it means mn is divisible by 5. But multiplication is commutative, so if mn is divisible by 5, then nm is also divisible by 5. Therefore, it is symmetric.
To check if it is transitive, we need to see if for any m, n, and p, if m~n and n~p, then m~p. If m~n, it means mn is divisible by 5, and if n~p, it means np is divisible by 5. Since multiplication is associative, the product of mn and np is (mn)(np) = m(np)(np). Since both mn and np are divisible by 5, their product m(np)(np) is also divisible by 5. Therefore, it is transitive.
Therefore, the relation m~n if and only if mn is divisible by 5 is an equivalence relation.
a) m~n if and only if mn is divisible by 5.
To check if it is reflexive, we need to see if for any m, m~m. If we choose m = 5, then 5 * 5 = 25, which is divisible by 5. Therefore, it is reflexive.
To check if it is symmetric, we need to see if for any m and n, if m~n, then n~m. If m~n, it means mn is divisible by 5. But multiplication is commutative, so if mn is divisible by 5, then nm is also divisible by 5. Therefore, it is symmetric.
To check if it is transitive, we need to see if for any m, n, and p, if m~n and n~p, then m~p. If m~n, it means mn is divisible by 5, and if n~p, it means np is divisible by 5. Since multiplication is associative, the product of mn and np is (mn)(np) = m(np)(np). Since both mn and np are divisible by 5, their product m(np)(np) is also divisible by 5. Therefore, it is transitive.
Therefore, the relation m~n if and only if mn is divisible by 5 is an equivalence relation.
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Consider the following relations in 1, then which of the following is an equivalence relation?a)m~n if and only if mn ≥ 0b)m~n if and only if mn > 0c)m~n if and only if |m| = |n|d)m~n if and only if mn ≤ 0Correct answer is option 'C'. Can you explain this answer?
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