Advanced High-School Mathematics

214 CHAPTER 4 Abstract Algebra

We display addition and multiplication on the integers modulo 5

in the following obvious tables:

+ 0 1 2 3 4

0 0 1 2 3 4

1 1 2 3 4 0

2 2 3 4 0 1

3 3 4 0 1 2

4 4 0 1 2 3

× 0 1 2 3 4

0 0 0 0 0 0

1 0 1 2 3 4

2 0 2 4 1 3

3 0 3 1 4 2

4 0 4 3 2 1

Exercises

1. Denote by 2Z⊆Zthe even integers. Is 2Zclosed under addition?
   Under multiplication?


2. Is the set ofoddintegers closed under either addition or multipli-
   cation?


3. On the setZof integers define the binary operation∗by setting
   x∗y=x+ 2y∈Z. Is the set of even integers closed under∗? Is
   the set of odd integers closed under∗?


4. LetU 2 (R)⊆Mat 2 (R) be defined by setting

U 2 (R) =













1 x

0 1







∣∣

∣∣

∣x∈R





.

IsU 2 (R) closed under matrix addition? Under matrix multiplica-

tion?

5. Let Xbe a set and let Sym(X) be the set of permutations ofX.
   Fix an element x ∈ X and show that Symx(X) is closed under
   function composition “◦.”


6. Let A be a set and let A ⊆ 2 A be the subset of the power set
   consisting of all finite subsets ofeven cardinality. Show that if
   |A| ≥3, thenA is not closed under either∩or ∪but itisclosed
   under +. (Why do we need to assume that|A|≥3?)