Chapter 3: Introduction to Sets Section 3.1: Operations on Sets Terms and Notation

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Abstract	Undefined Terms: Every definition uses terms other than the one being defined. If we insist that each of these terms also be defined, those definitions would introduce other terms whose definitions would, in turn, introduce even more terms. Eventually, we would find that our definitons have circled. In mathematics, we wish to avoid circularity, so certain basic terms are declared to be undefined. The concepts of point, plane, and set are examples of terms that we declare as undefined. Sets: Since " set " is, by choice, an undefined term, we give only an intuitive statement of what a set is. Thus, we understand a set to be a collection of objects. We insist, however, that a set, say A, satisfy the following two requirements: (1) The elements of A must come from some well understood universal set. For example, the set of all sets is not an acceptable universal set. (2) For each element x in the universal set, the sentence " x is an element of A " must be a statement; that is, the sentence must be true or false. For example, if n is a particular integer, the sentence " n is a large integer " is not a statement, so we will not permit the set A consisting of all large integers. Elements of a Set: Let A be a set with universal set U. If x is an element in U we write x ∈ A to mean that x is an element of A, and we write x ∈ A to denote that x is not an element of A. The empty set, denoted by the symbol ∅, is the set that contains no elements. Cardinality: If A is a set, the number of elements in A is called the cardinality of A and will be denoted by |A|. Set Notation: We use set braces {.. .} to enclose the elements of a set. For small finite sets, we can actually list the elements of the set. For example, if A = { 1, 2, 3 } then 1 ∈ A but 4 ∈ A. More commonly, we describe a set using set-builder notation wherein a set is defined using the form A = { x ∈ U | P (x) }. In this notation, the vertical line, " | " , is read " such that " and P (x) is a …
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