Bridge to Abstract Mathematics: Mathematical Proof and Structures

1.1 BASIC DEFINITIONS AND NOTATION 7

we pointed out that Ohio c$ B = (Massachusetts, Michigan, California} but
did not explicitly say that 5 4 B or that Harry Jones 4 B. The implicit
understanding was that, in the discussion involving this set B, the objects
under consideration, or the potential elements, were states, with the uni-
versal set U being a set of 50 elements.
The role then of a universal set is to put some bounds on the nature of
the objects that can be considered for membership in the sets involved in
a given situation.

SOME SPECIAL SETS

In mathematics the sets of greatest interest are those whose elements are
mathematical objects; included among these are sets whose elements are
numbers. In this context there are certain sets of numbers that serve as a
universal set so frequently that we assign them (widely used and recognized)
names and symbols.

DEFINITION 1

Throughout this text, we will denote by:

(a) N the set (1, 2, 3, 4,.. .) of all positive integers (natural numbers)

(b) Z the set {O, & 1, $2,... ). of all integers (signed whole numbers)

(c) Q the set of all rational numbers (quotients of integers)

(d) R the set of all real numbers (the reals)

(e) C the set of all complex numbers

These names are commonly used in the description of sets whose elements
are numbers. It is of vital importance, also, to realize that the universal set
specijied in the description of a set is as important as the rest of the definition.
For example, the set J = (x E Qlx2 2 2) is different from the set L = {x E
R I x2 > 21, even though the descriptions of both sets use thi same inequal-
ity (since, e.g., $ E L, but fi q! J). Considering these remarks, we may
streamline the notation used in our descriptions of some sets earlier, writing

and

We will study the sets N, Z, Q, R, and C as number systems in Chapters

9 and 10.

INTERVALS

Within the context of R as the universal set (the understanding throughout

most of elementary and intermediate calculus), there are other special sets,