Boolean Algebra Essentials

SYSTEMS

CHAPTER 1

SET THEORY

ESSENTIAL CONCEPTS

Sets and the Operations of Union, Intersection and Complement

Representation by Venn Diagrams

Fundamental Relations

Power and Universal Sets

Algebra of Sets

1.1 WHAT ARE SETS?

Definition of a Set

A set is a collection of objects. These may be numbers, people, or objects of any kind that are related by the fact that they belong to the set.

Definition of the Elements of a Set

The objects of a set are said to belong to the set, or to be members or elements of the set.

Notation of Sets and Their Elements

We denote sets and the objects in them by letters. If an object a belongs to a set S then we write

a ∈ S.

On the other hand if a is not an element of S then we write

a ∉ S.

Example of a Set

The collection of all natural numbers less than 10 is a set which may be denoted by N(10). The set has the elements

1,2,3,4,5,6,7,8,9.

Definition of Finite Sets

A set is a finite set if it has finitely many elements. N(10) of the above example is a finite set containing 9 elements.

Definition of Infinite Sets

A set is an infinite set if it has infinitely many members.

Example of the Set of All Natural Numbers

The set of all natural numbers, denoted by N, is an infinite set containing the natural numbers

1,2,3, ... , ....

Example of the Set of All Real Numbers

The set of all real numbers, denoted by R, is an infinite set containing the natural numbers,

1,2,3,...,

zero and the negative integers,

0, − 1, − 2, − 3, ... ,

the rational numbers

and the irrational numbers

π, √2, − e, ....

Notation For a Set In Terms of its Elements

If a set S has the elements a, b, c ... , then we sometimes write

S = {a, b, c, ... }.

Definition of a Singleton

A singleton is a set consisting of a single element.

Example of Singleton Sets

The set of all Presidents of the United States is a singleton consisting of a single element. Similarly the set

S = {√17}

is a singleton.

Definition of the Empty Set ø

The empty set ø is a set containing no elements.

Example of an Empty Set

The set whose elements are kings of the United States is an empty set since the United States has no kings.

Example of Complex Roots of x² + 1 = 0

The set S of complex roots of the equation

x² + 1 = 0

consists of the numbers ± i, (for i the imaginary constant whose square is − 1, i² = − 1):

S = {i, − i}.

Example of Real Roots of x² + 1 = 0

The set of real roots of the equation

x² + 1 = 0

is the empty set ɸ since there are no real roots to this equation.

Definition of Point Sets in 1, 2, and 3 Dimensions

A point set S on the real line

− ∞ < x < ∞,

the x, y plane

− ∞ < x, y < ∞,

or three dimensional space

− ∞ < x, y, z < ∞

is a set of points in the respective spaces.

Example of Some Point Sets

The point sets S, T, U, V of Figure 1.1 are defined respectively as

S = set of points x with − 4<x<2, x = 5, 6.5.

T = set of points (x, y) with

x² + y² = 1

U = set of points (x, y) with

x² / a² + y² / b² ≤ 1

V = set of points (x, y) with

|x + y| ≥ 1, x = y = 0.

Note that S consists of an interval of the real line together with the points 5, 6.5; T is the arc of the unit circle; U is the arc of an ellipse together with its interior, while V consists of the two infinite regions of Figure 1.1d together with the origin.

Figure 1.1a — The set S.

Figure 1.1b — The set T.

Figure 1.1c — The set U.

Figure 1.1d — The set V.

Defintion of Venn Diagrams

The representation of a set by points sets in the x, y plane as in Figure 1.2 is called a Venn Diagram.

Figure 1.2 — Venn Diagram.

1.2 INCLUSION RELATIONS BETWEEN SETS

Definition of Equality of Sets

Two sets A, B are equal,

A = B

if they consist of the same collections of objects.

Example of Some Equal and Unequal Sets

If A = {1, 2, 3 }, B = {1, 2, 3 }, C = {1, 2, 5} then

A = B, A ≠ C.