MTH 307 Homework #17 Jake Smith
11.2.2 Let A = {a, b, c, d, e}. Suppose R is an equivalence
relation on A. Suppose R has twoequivalence classes. Also aRd, bRc,
and eRd. Write out R as a set.
Answer: R = {(a, a), (b, b), (c, c), (d, d), (e, e), (a, d), (d,
a), (b, c), (c, b), (e, d), (d, e), (a, e), (e, a)}
11.2.4 Let {a, d, c, d, e}. Suppose R is an equivalence relation
on A. Suppose also that aRdand bRc, eRa and cRe. How many
equivalence classes does A have?
Answer: A has 1 equivalence class.
11.2.6 There are five different equivalence relations on the set
A = {a, b, c}. Describe the all.Diagrams will suffice.
Answer: R1 = {(a, a), (b, b), (c, c)}R2 = {(a, a), (a, b), (b,
a), (b, b), (c, c})R3 = {(a, a), (b, b)(b, c), (c, b), (c, c)}R4 =
{(a, a), (a, c), (c, a), (b, b), (c, c)}R5 = {(a, a), (a, b), (b,
a), (a, c), (c, a), (b, b), (b, c), (c, b), (c, c)}
11.2.8 Define a relation R on Z as xRy if and only if x2+y2 is
even. Prove R is an equivalencerelation. Describe its equivalence
classes.
Proof. (1) R is reflexive. Note that when xRx, x2 + x2 = 2x2
which is even.
(2) R is reflexive. The set of integers is closed over addition
so order does not matter
for addition and subtraction. Thus, x2 + y2 being true implies
the same for when their
order is reversed.
(3) R is transitive. Not sure how to prove it
transitive-ness
R has two equivalence classes: the even integers and the odd
integers.
11.3.2 List all of the partitions of the set A = {a, b,
c}.Answer: The partitions of A are:
{{a}, {b}, {c}}, {{a, b}, {c}}, {{a, c}, {b}}, {{a}, {b, c}},
{{a, b, c}}
11.4.4 Write the addition and multiplication tables for Z6.
+ [0] [1] [2] [3] [4] [5]
[0] [0] [1] [2] [3] [4] [5]
[1] [1] [2] [3] [4] [5] [0]
[2] [2] [3] [4] [5] [0] [1]
[3] [3] [4] [5] [0] [1] [2]
[4] [4] [5] [0] [1] [2] [3]
[5] [5] [0] [1] [2] [3] [4]
1
