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Equivalence Relations Homework

Directions: Write your homework legibly with a pencil. Complete the homework on time and by yourself. For each problem, write the instructions, label the solution/proof and show all steps. Do not turn in your scratch work. Staple your pages together, with the problems in the correct order, and use this page as a cover sheet with your name at the top right.

(1) For point and in a plane with rectangular coordinate system, let mean that Either show is an equivalence relation on or show that it is not.

(2) For point and in a plane with rectangular coordinate system, let means that either and Either show is an equivalence relation on or show that it is not.

(3) Define a relation on the set by if and only if for some   Either show is an equivalence relation on or show that it is not.

(4) Find all partitions of How many different equivalence relations are there on a three-element set?

(5) How many different equivalence relations are there on a four-element set?

(6) Let two complex numbers be equivalent if their real parts are equal. Prove that this is an equivalence relation and that the equivalence classes are in one-to-one correspondence with the real numbers.

(7) Let be the set of positive integers, and define on by if Prove that is an equivalence relation and that there is a one-to-one correspondence between the equivalence classes of and the set of all integers.

(8) Let be the set of nonzero integers, and define on by if Prove that is an equivalence relation and that there is a one-to-one correspondence between the equivalence classes of and the set of all rational numbers.

(9) For polynomials and with real coefficients, let mean Prove that is an equivalence relation and give the complete set of equivalence class representatives.

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