Let R be a binary relation on the set \mathbb{Z} . Which of the following definitions for R would make it an equivalence relation?

Question

Let   R  be a binary relation on the set   \mathbb{Z} . Which of the following definitions for   R  would make it an equivalence relation?

Answer

For all   x, y \in \mathbb{Z}, xRy  if and only if   x = -y .

Answer

For all   x, y \in \mathbb{Z}, xRy  if and only if   xy \ge 0 .

Answer

For all   x, y \in \mathbb{Z}, xRy  if and only if   x < y .

Answer

For all   x, y \in \mathbb{Z}, xRy  if and only if   x - y  is even.

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