Sets

Definition: A set S is a well defined collection of objects, called elements.

Here “well defined” means that any given object in the world at large (abstract or concrete) is either an element of S or it isn’t.

Note: S is a set x, is a proposition.

Representing sets:

1. As a list of elements: Note: the order of the elements doesn’t matter.

S= { 5, 6, 4, 2, 1, 9, 10, 10 } is the set with elements 1, 2, 4, 5, 6, 9, 10.

2. By a Venn diagram:

S 1 2

6 10

4 5 9

3. Like propositions, sets make sense in given contexts, called universal sets.

Given a context, a set can be represented by a truth table.

For example, if U= {1, 2, 3 ,4, 5, 6, 7, 8, 9, 10},

the truth table of the set S above:

x 1 2 3 4 5 6 7 8 9 10 x S

4. By a rule:

S={x{1,2,3,4,5,6,7,8,9,10}; }

Set up the context here

Write the rule here

Notation: x S

“x is an element of S” x S

Notation:

U=universal set

Notation: = The set of no elements.

Definition: Cardinality of a set = the number of elements of that set.

Examples: M= {, , , , }, then |M|=2 because in fact : M= {, }.

N={{ a,b,c}, {1,2}, 1, 2}, then |N|=

||=

|{}|=

|{{}}|=

|{, {,{}}}|=

Definition: Given two sets A and B, we say that A is a subset of

B or, equivalently, that A is included in B, if all the elements of A of A are also elements of B:

A B x, (xA xB).

We say A is strictly included in B if A B and AB.

Examples: For each of the following propositions, state whether it is true or false:

{1,2} {{1,2}, 1, 3}

{1,3} {{1,2}, 1, 3}

Note: A, A set.

Definition: The power set of A is the set whose elements are all

the subsets of A.

Examples: P( {1,2} )={, {1}, {2}, {1, 2}}

Notation:

|S|=cardinality

of set S

Notation:

A B: A is a subset of B.

A B: A is strictly

included in B.

Notation:

P(A): the power set of A.

P({1,2,3})={ }

P()=

P(P())=

Proposition: Let S be a set with |S|=n. Then P(S)=

Proof: Each subset A S is determined by its truth table:

x s1 s2 s3 s4 ... ... ... ... sn

xA

Operations with sets

Definitions: Let A and B be two sets.

The symmetric difference:

AB= (A-B) (B-A)

The difference:

A-B=A\B={x; xA xB}

The intersection of A and B:

AB={x; xA xB}

The union of A and B:

AB={x; xA xB}

Truth tables:

A B AB AB A-B B-A AB A’ B’ AB’

Examples:

1. A={x; x= the last digit of 4n for n }

B={x; x= the last digit of 6n+n5 n for n }

Find AB, AB, A-B, B-A, AB, AxB.

Application: Find all natural numbers m and n such that 4m =6n+n5 n.

Symmetric product:

AxB={(a,b); aA and bB} where (a,b)= an ordered pair of objects a and b.

“Ordered” means that (a,b)(b,a).

The complement of A:

A’={xU; xA}

Some set identities:

Proofs of set identities:

1. By logic propositions.

Example:

2. By truth tables.

Example:

A B C BC A(BC) AB AUC (AB) (AUC)

Short History

(A’)’=A.

(AB)’=A’B’ and (AB)’=A’B’.

A (BC)=(AB) (AUC). Distributivity of with respect to .

A (BC)=(AB) (AC). Distributivity of with respect to .

Georg Cantor (1845-1918): Introduced Set Theory, (1874).

Friedrich Ludwig Gottlob Frege (1848-1925):

“Naive” Set Theory, based on the assumption that

For any property P(x), there exists a set {x; P(x)} of all objects with that property.

Russell’s (and Zermelo’s) paradox:

Notes: In general, for any set X we find it natural that XX, but according to the “naive” set

theory rules, there actually can exist also sets X such that XX: for example, let X=the set of

all things which are not squares.

We note that the paradox is based on the self reference in the construction of R: XX.

But there is more to it than that. Imagine an immense truth table, each row and column

corresponding to a set.

column B

row A 1 if AB 0 if AB

Thus if we read all the 1’s and 0’s on the diagonal, we obtain the truth table of the set

{A; AA}. But if we change all 1’s into 0’s and 0’s into 1’s, we obtain the set

R={A; AA}. Ironically, this trick seems inspired by Cantor’s own diagonal argument: the

proof that the set of real numbers is not countable, of which we will speak later on.

Zermelo, Fraenkel: Axiomatic set theory.

Despite Zermelo, Fraenkel and others efforts to place arithmetic on stable foundations via

set theory: Trying to set up a minimum set of basic rules on which arithmetic should be

based.

Gödel's first incompleteness theorem:

For any consistent, effectively generated formal theory that contains basic arithmetic

truths, there is an arithmetical statement that is true, but not provable in the theory.

Very roughly, Gödel's goes like this: Consider all propositions about natural numbers:

P(x), where x denotes a number.

These propositions can be listed in some particular order. Consider a table whose rows

correspond to propositions Pi, and columns to natural numbers x:

Column x

Row Pn

“Pn(x) has a proof”

Now negate the propositions on the diagonal:

Pn(n) does not have a proof.

This is an aritmetical statement about the variable n, and as such it can be found in the list

of propositions Pi. Suppose it is the k-th proposition:

Pk(n): Pn(n) does not have a proof.

In particular, when n=k this means

Pk(k): Pk(k) does not have a proof.

Can we prove Pk(k)? But then it would be true that Pk(k) does not have a proof!

-Whitehead, Alfred North, and Bertrand Russell. Principia Mathematica, 3 vols,

Cambridge University Press, 1910, 1912, and 1913.

-Roger Penrose: “The emperor’s new clothes.”

-BBC documentary: dangerous knowledge:

http://www.youtube.com/watch?v=Cw-zNRNcF90

http://en.wikipedia.org/wiki/Axiomatic_set_theory#Axiomatic_set_theory