Sets, Membership and Extensionality

A set is a collection of any objects, considered as a single abstract object.

There is a set consisting of the earth, the sun, and the moon; another, consisting of the earth, the sun, the moon, and the planet Jupiter; and still another, consisting of the earth, the moon, number 8, and Bill Clinton. Thus, we can put into the same set objects whatsoever. Usually, we consider sets whose members are of the same kind: sets of people, sets of numbers, sets of sentences, etc. But this is not a restriction imposed by the concept of set; the theory allows us to form sets arbitrarily.

The objects that go into a set are said to be its members. And the basic relation on which set theory is founded is the membership relation; it holds between two objects just when the second object is a set and the first is a member of it.

The symbol for membership is ‘∈’ . It is employed as follows:

x∈ X

means that x is a member of the set X, and

x6∈ X

means that x is not a member of X.

Hence, if X is the set whose members are the earth, the moon, the number 8, and Bill Clinton, then:

Earth ∈ X, 8 ∈ X, Moon ∈ X, Clinton ∈ X, Nixon 6∈ X, 6 6∈ X, Jupiter 6∈ X, etc.

We also have: Clinton 6∈ Nixon, because Nixon is not a set.

Terminology: The membership symbol is occasionally used inside the English, e.g., ‘there is x ∈ X’ is read as: ‘there is a member, x, of X’. Similar self-explanatory phrases will be used throughout.

Sometimes (but not always!) ‘contains’, or ‘is contained’, means contains as a member, or is contained as a member. In these cases ‘X contains x’ means that x∈ X. We also say that x belongs to X, or that x is an element of X.

We use ‘x, y ∈ X’ as a shorthand for: ‘x ∈ X andy ∈ X’, and similarly for more than two members: ‘x, y, z ∈ X’.

Extensionality

A set is completely determined by its members. This means that sets that have the same members are the same set. Stated in full detail, the Extensionality Axiom says:

If X and Y are sets then X = Y iff every member of X is a member of Y and every member of Y is a member of X.

Note that the “only if” direction is trivial: X = Y means that X and Y are identical, hence they must have the same members. (This is actually a truth of first-order logic.) The real content of the axiom consists in the “if” direction: having the same members is sufficient for being the same set.

To see the implications of extensionality, consider the following two concepts: that of a human being and that of a featherless two-footed animal. In an obvious sense, the concepts differ.

But humans are featherless two-footed animals, and it so happens that there are no other such creatures besides humans. Hence, the set of humans is identical to the set of featherless two-footed animal. When forming sets, differences between concepts that cannot be cashed in terms of members are ignored.

The extensionality axiom provides the standard way of proving that sets are equal. If X and Y are sets, then to prove:

X = Y it suffices to show that, for every x,

x∈ X iff x∈ Y .

Ways of Denoting Sets

The simplest way of representing sets is by listing their members. The set is denoted by putting curly brackets, { }, around the list. The three examples given at the beginning of the section are denoted as:

{Earth, Sun, Moon}, {Earth, Sun, Moon, Jupiter}, {Earth, Moon, 8, Clinton}

The ordering of the list and repetitions in it do not matter:

{Earth, Moon, 8, Clinton} = {Clinton, Clinton, 8, Earth, Clinton, Earth, 8, Moon}

because every member of the left-hand side is a member of the right-hand side set, and every member of the right-hand side is a member of the left-hand side.

The method of listing the members is not practical when the list is too long, and not feasible if the set is infinite. Sometimes suggestive notations can be used for infinite sets, for example:

{0, 1, 2, . . .} or {0, 2, 4, . . .}

The first is set of all natural numbers (i.e., non-negative integers), the second–of all even natural numbers. But this method, which is based on guessing the intended rule, is very limited.

The most natural–and, in principle, perhaps the only–way of representing a set is by means of a defining condition: one that determines what objects belong to it. In English, the definition has the form:

the set of all ...

where ‘...’ expresses the condition in question. Thus, we have:

The set of all positive integers divisible by 7 or 9, the set of all planets, the set of all stars, the set of all atoms, the set of all USA citizens born in August 1991, the set of all British kings who died before 1940, and so on.

Note that finite listing can be seen as a special case of this kind of definition:

{earth, moon, 8, Clinton} = the set of all objects that are either the earth, or the moon, or number 8, or Clinton.

In mathematics the following is used:

{x : . . . x . . .}

It reads: The set of all x such that ...x... . Here, ‘...x...’ states the condition about x. Instead of ‘x’ any other letter can be used. We shall refer to it as the standard curly bracket notation.

The examples given above can be therefore written as follows:

{x : x is a positive integer divisible by 7 or 9}, {x : x is a planet },

{v : v is a star}, {y : y is an atom}, {z : z is a USA citizen born in August 1991}, {x : x is a British king who died before 1940}, and so on.

This is not to say that every set can be denoted by an expression of the last given form, or–for that matter–by some other expression. In mathematics we allow for the possibility of sets not denoted by any expression in our language; just as there may be atoms that no description can pick.

Variants of the Notation: Usually, set members are chosen from some fixed given domain (itself a set). If U is the domain in question, then the set of all members, x, of U that satisfy ...x... is, of course:

{x : x ∈ U and . . . x . . .}

An alternative notation is:

{x ∈ U : . . . x . . .}

which reads: ‘the set of all x in U such that ...x...’ . Thus, if N is the set of all natural numbers, then:

{x ∈ N : x + 1 is divisible by 3} = {x : x ∈ N and x + 1 is divisible by 3}

Occasionally, the domain in question is to be understood from the context. It is also customary to employ variables that range over fixed domains. If in the last example it is understood that ‘x’ ranges over the natural numbers, then we can omit the reference to N and write simply

{x : x + 1 is divisible by 3}

Other variants of the notation involve the use of functions. For example,

{2x : x ∈ N} and {x² : x∈ N}

are, respectively, the set of all numbers of the form 2x and the set of all numbers of the form x², where x ranges over N (i.e., the set of all even natural numbers and the set of all squares).

We can use for these sets the standard notation; but this would result in longer expressions.

For example:

{x² : x∈ N} = {z : there is x ∈ N, such that z = x²}

Once you get used to them you will find these and other notations self-explanatory. The following exercises will help you to get accustomed to set-theoretic notations and phrasings.

Homework 5.1 Translate the following into the standard curly-bracket notation.

(1) The set of all people who like themselves.

(2) The set of all integers that are smaller than their squares.

(Recall, the square of x is x².)

(3) The set of all people married to 1992 Columbia students.

Rewrite the following in the curly-bracket functional notation. You can use ‘N ’ and ‘Z’ to denote, respectively, the set of natural numbers and the set of integers. For (6) use ‘father(x)’

to denote the father of x.

(4) The set of all positive multiples of 4.

(5) The set of all successors of integers divisible by 5.

(6) The set of all fathers of identical twins.

Describe in English the following sets, use short, neat descriptions. (‘ Livings ’, ‘ Humans ’, and ‘ Planets ’ have the obvious meanings.)

(7) {x ∈ Livings : x has two legs}

(8) {x ∈ Humans : x has more than one child}

(9) {x ∈ Planets : x is larger than the earth}

Rewrite the following in the standard curly-bracket notation.

(10) {3x : x ∈ Primes}

(11) {x − y : x ∈ Primes, y ∈ Primes}

(12) {2x + y² : x∈ N, y ∈ Primes}

Note: The concept of a set is primitive. It cannot be defined by reduction to more basic concepts. Explanations and examples (like the ones just given) may serve to get the concept

across, but they do not amount to definitions. In an indirect way, the concept is determined by what we take to be the basic properties of sets. The same takes place in Euclidean geometry, where the undefined concepts of point, line and plane are indirectly determined by the geometrical axioms. Like geometry, set theory is a system based on axioms. Some are “obvious”. Others, belonging to more sophisticated parts of the theory, require deep understanding. Except for extensionality, the axioms are not discussed here.

Singletons The set {x} has a single member, namely, x. Such a set is called a singleton, or a unit set; {x} is the singleton of x, or the unit set of x.

One may be tempted to identify the singleton of x with x itself. The temptation should be resisted. The singleton {Clinton} is a set containing Clinton as its sole member. Clinton himself is a man, not a set. Just so, one distinguishes between John the man and the one-member committee having John as its only one-member. If all the committee one-members except John perish in a crash, the committee becomes a one-member committee; but you do not want to say that it becomes a man. The standard version of set theory has an axiom, called the regularity axiom, which implies that nothing can be a member of itself. It therefore implies that, for all x, x6= {x} (because x ∈ {x}, but x 6∈ x).

The singleton of x is {x}, the singleton of {x} is {{x}}, the singleton of {{x}} is {{{x}}}, and so on: {. . . {{x}} . . .}. It can be shown (assuming the regularity axiom) that all of these are different from each other.

The Empty Set Among sets we include the so-called empty set: one that has no members.

At first glance one may find this strange, as one might find strange, at first, the idea of the number zero. In fact, the concept is simple, highly useful and easily handled.

We speak about the empty set, because there is only one. This follows from extensionality: If X and X⁰ are sets that have no members, then X = X⁰, because they have the same members (for every x: x∈ X iff x ∈ X⁰). The empty set is denoted as:

∅ .

Note that every object that is not a set (e.g., every physical object) has no members. Ex-tensionality does not make these objects equal to ∅, because extensionality applies only to sets.