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Consider the following relations, 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 'A,C'. Can you explain this answer?
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Consider the following relations,which of the following is an equivale...
The equivalence class of  α under ~, denoted [α] [, is defined as  [α] = { x ∈ X | x ~ α }
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Consider the following relations,which of the following is an equivale...
The relation m~n if and only if mn is an equivalence relation because it satisfies the three properties of an equivalence relation:

1) Reflexivity: For any element m, m~m because mm = m^2, which is true for all values of m.

2) Symmetry: For any elements m and n, if m~n then n~m. This is true because mn = nm, so if mn is true, then nm is also true.

3) Transitivity: For any elements m, n, and p, if m~n and n~p then m~p. This is true because if mn = np and np = pq, then mn = pq, so m~p.

Therefore, the relation m~n if and only if mn is an equivalence relation.
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Consider the following relations,which of the following is an equivale...
For a take (1,0),(0,-1) and for d take (1,0),(0,1) so transitivity fails
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Consider the following relations,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 'A,C'. Can you explain this answer?
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