correspon- dence (bijection) between R* and the set of all equivalence classes for this equivalence relation. 11. Let A be a nonempty set and let ~ be an equivalence relation on A. Prove each of the...

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Extracted text: correspon- dence (bijection) between R* and the set of all equivalence classes for this equivalence relation. 11. Let A be a nonempty set and let ~ be an equivalence relation on A. Prove each of the following: (a) For each a. bE 4, ax b if and only if [a] n [b] = 0. (b) For each a. b e A, if (a] 7 [b], then 2e] n ( = . (c) For each a b € A, if la] N [b] 0 then la] = b| Explorations and Activities 12. A Partition Defines an Equivalence Relation. Let 4 = fa b.c d, e} and Tet € = {{a, b. c} {d_e}} fa) Explain why C is a partition of A. Define a relation s on 4as follone Fora