Bridge to Abstract Mathematics: Mathematical Proof and Structures

1.1 BASIC DEFINITIONS AND NOTATION 13

POWER SET

We suggested earlier that sets can themselves be elements of sets. One sit-

uation in which this always happens arises in the following definition.

DEFINITION 5

Let A be a set. We denote by 9(A), the power set of A, the set of all subsets of A.

EXAMPLE 5 Discuss the structure of 9(A), where A = (1,2, 3,4).

Discussion The elements of 9(A) are precisely the subsets of A. So, for

instance, the set (2,4) is an element of 9(A). Also (11 E 9(A). Since

A E A, then A = (1,2,3,4) E 9(A). You should list 13 other elements of

9(A). In the event that your list falls one short of the number we've

specified, perhaps you have not considered the question, "is @ an element

of P(A)?" The answer clearly depends on an issue we raised, but did not

settle, earlier, namely, whether 0 G A. You may want to look ahead

to Article 4.1 (Example 1) for the final resolution of that problem.

Other questions about 9(A) might easily be raised. For instance, is

1 E 9(A)? Is (2) G 9(A)? Is (01 c_ 9(A)? You should settle these ques-

tions and formulate similar ones.

Exercises

1. Express these sets via the roster method; that is list the elements in each:
   A= {xERIx~+x~- 12~=0)
   B= {x~R13/(x+ 1)+3/(x2+x)= -2)
   C={XEN~-~~<X<'Z)
   D = {x E N 1 x is prime and divisible by 2)
   E={xEZ~-~<X<~}
   F = {x(x was or is a Republican president of the United States}
   G = {x lx is a planet in the Earth's solar system}
   H = {xlx is a month of the year)
   I = {a E R I f(x) = x/(x2 - 3x + 2) is discontinuous at x = a}
   J = {a E R I f(x) = 1x1 fails to have a derivative at x = a)
   K = {a E RI f(x) = 3x4 + 4x3 - 12x2 has a relative maximum at x = a)
   L = {x~Rl(x + 2)'12 =(7 - x)'l2 - 3)
   M={x~R112x~+2x- lI=Ix2-4x-61)
   N = {zEC~Z~ = -1)
   O = (zECJZ~ = I}


2. Express the following sets by either interval notation (including (a) or one of the
   symbols N, Z, Q, R, or C:

*(a) A = {XE Rlsin 2x= 2sin xcosx)