1. Let P be a partition of the nonempty set A. For x, y E A, define xQy if and only if there exists C E P such that x E C and y E C. Prove that (a) Q is an equivalence relation on A. (b) A/Q = P. 2....

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Extracted text: 1. Let P be a partition of the nonempty set A. For x, y E A, define xQy if and only if there exists C E P such that x E C and y E C. Prove that (a) Q is an equivalence relation on A. (b) A/Q = P. 2. Let m e N, then show that (a) For integers x, y, x = y(mod m) if and only if the remainder when x is divide by m equals to remainder when y is divided by m. (b) Zm consists of m distinct equivalence classes: Zm = {0,1, 2, . ., m – 1}. ...