In each item below, verify that ~ is an equivalence relation on the given set. (a) Let S = Z × N. Define the relation ~ on S as: (a, b) ~ (c, d) if and only if ad – bc = 0. Moreover, if a and b are...

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Extracted text: In each item below, verify that ~ is an equivalence relation on the given set. (a) Let S = Z × N. Define the relation ~ on S as: (a, b) ~ (c, d) if and only if ad – bc = 0. Moreover, if a and b are relatively prime, how are c and d related to a and b? (b) Let ~ be a relation defined on Z by a Rb if and only if 2a+b = 0 (mod 3). Determine the distinct equivalence classes determined by this equivalence relation. (c) Let ~ be a relation defined on Z by a R b if and only if a² = b³ (mod 4). Determine the distinct equivalence classes determined by this equivalence relation. (Hint: There are only three distinct such congruence classes.)