1.1 INTRODUCTION TO SETS
(1) A set is a collection of items. These items are called the elements or members of
the set. Notice that we put elements in curly brackets to distinguish sets from
other objects. Usually we let upper-case letters, such as A and B, denote sets.
(2) When all the elements of the set are written out, we refer to this as roster notation; when we define the set in term of its properties, we refer to this as set-builder
notation.
(3) If a is an element of a set S, then we can use the notation a ∈ S, otherwise, we
can use the notation a ∈
/ S.
(4) If every element of a set A is also an element of another set B, we say that A is
a subset of B and denote as A ⊆ B. Otherwise, we say that A is not a subset of
B, and write as A * B.
(5) If A ⊆ B and there is at least one element of B is not an element of A, then A
is a proper subset of B and we write A ⊂ B (The only non-proper subset of B is
itself). These notations are similar as < and 6.
(6) The empty set, witten as ∅ or {}, is the set with no element.
(7) The universal set is the set of all elements being considered and is denoted by U .
(8) A Venn diagram is a way of visualizing set. The universal set is represented by a
rectangle and sets are represented as circles inside the universal set.
U
B
A
(9) Set Operation
Given a universal set U and a set A ⊆ U , the complement of A, written as Ac is
all elements that are contained in the universal set U , but not in A, that is
Ac = {x|x ∈ U, x ∈
/ A}
.
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1.1 INTRODUCTION TO SETS
AcomplementVenn.eps
The complement is equivalent to negation in the logic chapter.
The union of two sets A and B, written as A ∪ B, is the set of all elements that
belong to A and B, or to both. Thus
A ∪ B = {x|x ∈ A or x ∈ B}
2circleVenn.eps
The set union is equivalent to inclusive disjunction in the logic chapter.
The intersection of two sets A and B, written as A∩B, is the set of all elements
that belong to both the set A and to the set B. Thus,
A ∩ B = {x|x ∈ A and x ∈ B}
1.1 INTRODUCTION TO SETS
3
2circleVenn.eps
The set intersection is equivalent to the conjunction in the logic chapter.
Guchao Zeng; Department of Mathematics, Texas A&M University, College Station, TX
77843, USA; [email protected]