Math 134 Honors Calculus Fall 2016
Handout 4: Sets
All of mathematics uses set theory as an underlying foundation. Intuitively, a set is a collection of objects,
considered as a whole. The objects that make up the set are called its elements or its members. The
elements of a set may be any objects whatsoever, but for our purposes, they will usually be mathematical
objects such as numbers, functions, or other sets. The notation x∈Xmeans that the object xis an element
of the set X. The words collection and family are synonyms for set.
In rigorous axiomatic developments of set theory, the words set and element are taken as primitive undefined
terms. (It would be very difficult to define the word “set” without using some word such as “collection,”
which is essentially a synonym for “set.”) Instead of giving a general mathematical definition of what it
means to be a set, or for an object to be an element of a set, mathematicians characterize each particular set
by giving a precise definition of what it means for an object to be a element of that set—this is called the set’s
membership criterion. The membership criterion for a set Xis a statement of the form “x∈X⇔P(x),”
where P(x) is some sentence that is true precisely for those objects xthat are elements of X, and no others.
For example, if Qis the set of all rational numbers, then the membership criterion for Qmight be expressed
as follows:
x∈Q⇔x=p/q for some integers pand qwith q6= 0.
The essential characteristic of sets is that two sets are equal if and only if they have the same elements. Thus
if Xand Yare sets, then X=Yif and only if every element of Xis an element of Y, and every element of
Yis an element of X. Symbolically,
X=Yif and only if ∀x, x ∈X⇔x∈Y.
If Xand Yare sets such that every element of Xis also an element of Y, then we say Xis a subset of Y,
written X⊆Y. Thus
X⊆Yif and only if ∀x, x ∈X⇒x∈Y.
The notation Y⊇X(“Yis a superset of X”) means the same as X⊆Y. Using the concept of subsets,
we can restate the criterion for two sets to be equal as follows:
X=Yif and only if X⊆Yand Y⊆X.
If X⊆Ybut X6=Y, we say that Xis a proper subset of Y(or Yis a proper superset of X). Some
authors use the notations X⊆Yand Y⊇Xto mean that Xis a proper subset of Y; however, since
other authors use the symbol “⊆” to mean any subset, not necessarily proper, we generally avoid using this
notation, and instead say explicitly when a subset is proper.
Defining Sets
We already know about the set Rof all real numbers, and its subset R+of all positive real numbers. Most
other sets we deal with will be built up from these in various ways. There are just a few ways to define new
sets. You will see that in each case, the set is completely determined by its membership criterion.
•Defining a set by listing elements: Given any list of objects that can be explicitly named, the set
containing those objects and no others is denoted by listing the objects between braces: {c1, c2,...,cn}.
The membership criterion is easy to express:
a∈ {c1, c2,...,cn} ⇔ a=c1or a=c2or ... or a=cn.
For example, the set {0,1,2}contains the numbers 0, 1, and 2, and nothing else. Because a set is
completely determined by which elements it contains, it does not matter what order the elements are
listed in or whether they are repeated; the notations {0,1,2},{2,1,0}, and {0,0,1,2,1,1}all denote
the same set. A set containing exactly one element, such as {1}, is called a singleton. The set
containing no elements at all is called the empty set; it is usually denoted by ∅, but it can also be
denoted by { }.
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