Advanced High-School Mathematics

190 CHAPTER 4 Abstract Algebra

Determine which of the following real numbers are inQ(2):

π,

2

3

,

10

2

, cos(π/4), 12 ,

3

4

, 12 π,

π

3

2. True or false: Z⊆Q(p)for any prime numberp.


3. Consider the setS={ 1 , 2 , 3 , ..., 10 }. Define the sets

A = {subsetsT ⊆S| |T|= 2}

B = {subsetsT ⊆S||T|= 2, and ifx,y∈T then|x−y|≥ 2 }

Compute|A|and|B|.

4. Given the real number x, denote by [x] the largest integern not
   exceedingx. Therefore, we have, for example, that [4.3] = 4, [π] =
   3 , [e] = 2, [−π] =− 4 ,and

[ 10

3

]

= 3. Define the setAof integers

by setting

A =









^1

2

100



,



^2

2

100



,



^3

2

100



, ...,



^99

2

100



,



^100

2

100











and compute|A|.

4.1.2 Elementary operations on subsets of a given set

Let A andB be subsets of some bigger setU (sometimes called the
universal set; note thatU shall just determine a context for the en-
suing constructions). We have the familiarunion andintersection,
respectively, of these subsets:

A∪B = {u∈U|u∈Aor u∈B},and

A∩B = {u∈U|u∈Aandu∈B}.