For the following relations determine whether they are equivalence relations and

Question

For the following relations determine whether they are equivalence relations and prove your answer (that is, show that all axioms hold, or indicate which one fails). For the set P of all people on Earth let (a) P_1 ~ P_2 means P_1 looks like P_2; (b) P_1 ~ P_2 means P_1 has the same biological parents as P_2; (c) P_1 ~ P_2 means P_1 has the same at least one biological parent as P_2; (d) S_1 ~ S_2 means S_1 has the same major as S_2 (here S_i denotes a junior student in a college, only one major is allowed); (e) P_1 ~ P_2 means P_1 tolerates the presence of P_2. In cases when ~ is an equivalence relation try to describe as short as possible partition P/~.

Explanation / Answer

(a) and (b) are satistify the reflexive,symmetric and transitivity properties .Hence (a) and (b) are equivalence relation.

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