Math 220 Assignment 8
Due November 18th
1. Define the relation R on Z by aRb if and only if a2 − b2 is divisible by 5.
(a) Prove that R is an equivalence relation.
(b) What are the equivalence classes of R? Prove your answer.
2. Let ∗ be an associative binary operator on A, with an identity element e.
Let S ⊆ A, and suppose S is closed under ∗. Define the relation R by aRb
if and only if there exists an element c ∈ S such that b = a ∗ c. Prove that
R is an equivalence relation.
3. Let A be a set, and R be a reflexive and transitive relation on A.
(a) Define ' on A as follows: for a, b ∈ A, a ' b if and only if aRb and
bRa. Show that ' is an equivalence relation.
(b) Let C be the set of equivalence classes of A under '. Define ≤ on
C as follows: for equivalence classes [a] and [b], [a] ≤ [b] if and only
if there exists a0 ∈ [a], b0 ∈ [b], such that a0 Rb0 . Show that ≤ is a
partial order.
4. This problem is about constructing Q. Let A ⊂ Z × Z be set {(a, b) :
a, b ∈ Z, b 6= 0}. Define the relation ' on A by (a, b) ' (c, d) if and only
if ad = bc.
(a) Prove that ' is an equivalence relation.
(b) Define a binary operator ∗ on A by (a, b) ∗ (c, d) = (ac, bd). Prove
that, if (a, b) ' (a0 , b0 ) and (c, d) ' (c0 , d0 ), then (a, b) ∗ (c, d) '
(a0 , b0 ) ∗ (c0 , d0 ).
(c) Show that, if a 6= 0, then (a, b) ∗ (b, a) ' (1, 1).
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