Homework Assignment # 5
Math 248
Kaul
Spring 2014
Due Tuesday, May 20
Instructions: To receive full credit, each solution must be neat and legible. Explain your reasoning
fully and use complete sentences when appropriate – an answer without an explanation will receive
no credit. Staple the homework sheet to the front of your work.
1. Let A = {1, 2, 3, 4}. Give an example of a relation on A that is
(a) reflexive and symmetric, but not transitive.
(b) reflexive and transitive, but not symmetric.
(c) symmetric, but neither reflexive nor transitive.
(d) transitive, but neither reflexive nor symmetric.
2. Define a relation R on Z by x R y if and only if xy ≥ 0. Is R reflexive? symmetric? transitive?
3. Determine whether the relation R defined on N by a R b if and only if a2 + b2 is even is an
equivalence relation.
4. Let R and S be equivalence relations on the nonempty set A. Prove or disprove:
(a) R ∪ S is an equivalence relation on A.
(b) R ∩ S is an equivalence relation on A.
5. Define a relation ∼ on Z by a ∼ b if and only if a3 = b3 .
(a) Prove that ∼ is an equivalence relation on Z.
(b) Identify the distinct equivalence classes of Z/ ∼
(describe the classes algebraically).
6. Define the relation ∼ on Z by a ∼ b if and only if 5a ≡ 2b mod 3.
(a) Prove that ∼ is an equivalence relation on Z.
(b) Identify the distinct equivalence classes of Z/ ∼
(give a complete list of the classes, and make sure to explain why your list is complete).
7. Define the relation ∼ on R × R by (a, b) ∼ (c, d) if and only if 3(c − a) = d − b
(a) Prove that ∼ is an equivalence relation on R × R.
(b) Identify the distinct equivalence classes of R × R/ ∼
(describe the classes geometrically).
8. For each α ∈ R let Aα = {(x, y) ∈ R × R | y = α − x2 }.
(a) Prove that A = {Aα | α ∈ R} is a partition of R × R.
(b) Describe the equivalence relation induced by A .
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