MATH 531 University of New Hampshire Distinct Equivalence Classes Questions

Due Wednesday 11/18/2020

Instructions:

Write your name and Math 531.03 HW5 in the upper-right corner of page 1.

Solve the following problems. Things to be proven should be written as proofs.

Copy out the statement of problems before solving them.

Write proofs in complete, grammatical sentences, and in easily legible handwriting.

Scan and upload your homework to Canvas as a single PDF document.

If you take pictures with a phone, be sure to photograph in good light with no shadows

across the page, with the camera directly above the page (not at an angle), and hold the

camera still to prevent blurring. Make sure your work is easily readable, and if not, try

again. Combine the photos into a single PDF using a phone scanner app.

Problem 1. Let A = {1, 2, 3, 4, 5, 6}. The distinct equivalence classes resulting from an equiv-

alence relation R on A are {1, 4, 5}, {2, 6}, and {3}. What is R?

Problem 2. Let R be a relation on Z defined by a R b if a + b is even. Prove that R is an

equivalence relation.

Problem 3. Suppose even in problem 2 is changed to odd. Is R still an equivalence

relation? Which of the properties reflexive, symmetric, and transitive does this new R have?

Problem 4. Let S be a nonempty subset of Z and let R = {(x, y) ? S × S : 3 | (x + 2y)}.

(a) Prove that R is an equivalence relation on S.

(b) What are the distinct equivalence classes of R for S = {?7, ?6, ?2, 0, 1, 4, 5, 7}.

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