Sets¶

Sets are collections of objects of some sort or another. Those objects can be numbers, vectors, houses, or even other sets. However, the sets encountered most frequently are sets of numbers.

The foundations of almost all of mathematics is now built upon sets.

Definition 10 (Set)

\(\def\set#1{\mathcal{#1}}\) A set, \(\set{A}\) is a mathematical object which is defined by a binary relationship, \(\in\), between the \(\set{A}\) and any other object \(o\).

If the object \(o\) is an element in the set (a member of the set) then \(o \in \set{A}\).

Subsets¶

The notion of a set in its own right does not provide us with many useful tools for understanding it or its structure.

We can introduce the concept of a “subset” to try and build this.

Definition 11 (Subset)

Let \(\set{A}\) and \(\set{B}\) be sets.
Let \(b \in \set{B}\) be some element from \(\set{B}\).

If \(b \in \set{A}\) for every element \(b \in \set{B}\) then \(\set{B}\) is a subset of \(\set{A}\).

We can write this as \(\set{B} \subset \set{A}\).

Set operations¶

With sets and subsets defined, it is possible to start considering operations which might exist between two sets.

Definition 12 (Set Union)

Let \(\set{A}\) and \(\set{B}\) be sets.

The union, \(\set{A} \cup \set{B}\), of the sets is the set which contains all of the elements contained in both \(\set{A}\) and \(\set{B}\).

Definition 13 (Set intersection)

Let \(\set{A}\) and \(\set{B}\) be sets.

The intersection, \(\set{A} \cap \set{B}\) of the sets is the set which contains all of the elements which are both members of \(\set{A}\) and \(\set{B}\). That is \(x \in \set{A} \cap \set{B}\) if and only if \(x \in \set{A}\) and \(x \in \set{B}\).

Definition 14 (Set difference)

Definition 15 (Power set)

Ordering¶

Ordering is a property which can apply to a set to provide the intuitive quality of a sequence to the elements of a set.

Ordering is defined by a relationship on the elements of a set which dictate that order, and describe the concepts of greater and lesser than elements of a set.

Definition 16 (Partial order)

A binary relation, \(\leq\), on a set, \(\set{A}\) is called a partial order if and only if:

\(\leq\) is reflective (\(x \leq x \forall x \in \set{A}\))
\(\leq\) is symmetrical (if \(x \leq y\) and \(y \leq x\) then \(x = y\) for all \(x,y \in \set{A}\)).
\(\leq\) is transitive (if \(x \leq y\) and \(y \leq z\) then \(x \leq z\) for \(x, y, z \in \set{A}\)).

With the notion of ordering defined we can extend the definition of a set with it.

Definition 17 (Partially-ordered set)

A set which has a partial ordering relationship defined on it is called a partially-ordered set.

Bounds¶

Definition 18 (Bounding)

A partially-ordered set \(A \subset B\) is bounded above if there is a number \(b \in B\) such that \(a \leq b\) for all \(a \in A\). The number \(b\) is the upper bound for \(A\).

A partially-ordered set \(A \subset B\) is bounded below if there is a number \(b \in B\) such that \(a \geq b\) for all \(a \in A\). The number \(b\) is the lower bound for \(A\).

Definition 19 (Supremum)

The supremum, or least upper bound, of a partially-ordered set \(A \subset B\) is the number \(s\) which meets the criteria:

\(s\) is an upper bound of \(A\)
if \(b\) is any upper bound for \(A\) then \(s \leq b\).