SAM I
Sets
Lecture 1
Contents
Equality
Empty set
Pairing
SAM I
Sets
Singletons
Unions
Powersets
Lecture 1
Separation
SAM — Seminar in Abstract Mathematics [Version 20130219]
is created by Zurab Janelidze at Mathematics Division, Stellenbosch Univeristy
SAM I
Sets
Lecture 1
1 Equality
Contents
Equality
2 Empty set
Empty set
Pairing
3 Pairing
Singletons
Unions
4 Singletons
Powersets
Separation
5 Unions
6 Powersets
7 Separation
Elements and equality of sets
SAM I
Sets
Lecture 1
Contents
Equality
Empty set
In the language of sets variables represent objects called sets.
We have a binary relational symbol ∈. We read x ∈ y as x is
an element of y . Intuitively, a set is a collection of objects
(which are themselves sets) called elements of the set.
Axiom (equality of sets)
Pairing
Singletons
Unions
Two sets X and Y are equal if and only if they have same
elements:
Powersets
Separation
X =Y
⇔
∀z (z ∈ X ⇔ z ∈ Y )
Equality is reflexive, symmetric, and transitive
Prove that for any sets X , Y , Z we have X = X ,
X = Y ⇒ Y = X , and (X = Y ∧ Y = Z ) ⇒ Y = X .
Empty set
SAM I
Sets
Lecture 1
We write ∅ for the empty set.
Contents
Equality
Empty set
Pairing
Singletons
Unions
Powersets
Axiom (empty set)
For any set x we have x ∈
/ ∅.
Empty set is unique
Show that for any set X if
Separation
∀x (x ∈
/ X)
then X = ∅.
Pairing
SAM I
Sets
Lecture 1
Contents
Equality
Empty set
Pairing
Singletons
In the language of sets, for any two sets x and y we write
{x, y } to represent a set whose only elements are x and y .
This is expressed by the following axiom:
Axiom (pairing)
For any three sets x, y and z we have:
Unions
Powersets
z ∈ {x, y }
⇔
(z = x ∨ z = y )
Separation
Elements in {x, y } are not ordered
Prove that {x, y } = {y , x} for any two sets x and y .
Singletons
SAM I
Sets
Lecture 1
Contents
Equality
Definition
A singleton is a set of the form {x, x}, which is abbreviated as
{x}.
Empty set
Pairing
Singletons
Singletons
Unions
Prove that for any two sets x, y we have:
Powersets
Separation
x ∈ {y }
A singleton is nonempty
Prove that {∅} =
6 ∅.
⇔
x = y.
Unions
SAM I
Sets
Lecture 1
Contents
For any set X we write
elements of X .
S
X to represent the union of all
Axiom (union)
Equality
Empty set
Pairing
Singletons
For any two sets x, X we have:
[
x∈
X ⇔ ∃A (x ∈ A ∧ A ∈ X ).
Unions
Powersets
Separation
Empty union is empty
S
Show that ∅ = ∅.
Another empty union
S
Show that {∅} = ∅.
Powersets
SAM I
Sets
Lecture 1
Contents
Equality
Empty set
Definition
We say that a set X is a subset of a set Y , and write X ⊆ Y ,
when
∀z (z ∈ X ⇒ z ∈ Y ).
Pairing
Singletons
Unions
Powersets
Separation
For any set X we write P(X ) to represent the set of all subsets
of X , called the power set of X .
Axiom (powerset)
For any two sets X , Y we have:
X ∈ P(Y )
⇔
(X ⊆ Y ).
Separation
SAM I
Sets
Lecture 1
Contents
Equality
Empty set
Pairing
For any formula α(x, y ), and for any two sets X , Y we write
{y ∈ Y | α(X , y )} to represent the subset of Y consisting of
those elements y which satisfy α(X , y ). This is expressed by
the following axiom:
Axiom schema (separation)
Singletons
Unions
For any three sets X , Y , x we have:
Powersets
Separation
x ∈ {y ∈ Y | α(X , y )}
⇔
(x ∈ Y ∧ α(X , x)).
Exercise
Show that P({∅}) = {x ∈ P({∅}) | x =
S
x}.