Advanced High-School Mathematics

SECTION 4.1 Basics of Set Theory 195

A∩(B+C) = (A∩B) + (A∩C), whereA, B, C⊆U

A+ (B∩C) = (A+B)∩(A+C), whereA, B, C ⊆U.

11. Letpbe a fixed prime and letQ(p)be the set defined in Exercise 1
    of Subsection 4.1.1. Interpret and prove the statement that

⋂

all primesp

Q(p) = Z.

12. Interpret and prove the statements

(i)

⋂∞

n=1

(

[0,^1 n]

)

= { 0 }

(ii)

∞⋂

n=1

(

]0,n^1 ]

)

= ∅

4.1.3 Elementary constructions—new sets from old

We have already encountered an elementary construction on a given
set: that of thepower set. That is, if S is a set, then 2S is the set
of all subsets of the set S. Furthermore, we saw in the theorem on
page 189 that ifSis a finite set containingnelements, then the power
set 2S contains 2n elements (which motivates the notation in the first
place!). Next, letAandB be sets. We form theCartesian product
A×Bto be the set of allordered pairsof elements (a,b) formed by
elements ofAandB, respectively. More formally,

A×B = {(a,b)|a∈Aandb∈B}.

From the above, we see that we can regard the Cartesian planeR^2
as the Cartesian product of the real lineRwith itself: R^2 =R×R.
Similarly, Cartesian 3-spaceR^3 is justR×R×R.

Here are a couple of constructions to think about. Perhaps you
can see how a right circular cylinder of heighth and radiusr can be
regarded asS×[0,h], where S is a circle of radiush. Next, can you