Download Exercise sheet 5

Algebraic Structures (MST20010)
2016/2017
Problem sheet 5
1. Write the addition and multiplication tables of Z/4Z.
2. (a) You have 7 pieces of paper, and you apply the following procedure
as many times as you want: Pick any one of your pieces of paper
and cut it in 7.
Show that you can never get 1997 pieces of paper. Hint: Think
modulo 6.
(b) Find the remainder in the division by 3 of each of the following
numbers (i.e. compute them in Z/3Z):
892 ,
(1315 · 526 ) + (4 · 2632 ).
3. Let S be a set. A partition of S is a collection of non-empty sets Ai ,
for i in some index set I, such that every element of S belongs to one
of the Ai and Ai ∩ Aj = ∅ whenever i 6= j. If I has k elements we say
that the Ai form a partition of S into k subsets.
Example: {1, 3}, {2} forms a partition of {1, 2, 3} into 2 subsets.
(a) Give an example of a partition of {1, 2, 3, 4, 5} into 2 subsets and
of one of {1, 2, 3, 4, 5} into 3 subsets.
(b) Let ∼ be an equivalence relation on a set S, and let Ai (for i in
some index set I) be the list of all the different equivalence classes
in S for the relation ∼. Show that the Ai form a partition of S.
(c) Conversely, assume that the sets Ai (again for i in some index set
I) form a partition of S, and define x ∼ y if and only if x and y
belong to the same set Ai .
i. Show that ∼ is an equivalence relation on S.
ii. Show that the equivalence classes of ∼ are exactly the sets
Ai .
Observe that it shows that the equivalence relations on S correspond
to the partitions of S.
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