Let
A
and
B
be any two sets defined by
A = { 3, 4 }
B = { 3, 4, 5 }
Every member of A is also a member of B.
We say that
A is included in B or,
A is a subset of B.
Definition 4.2 (Subset)

A set
S
is included in another set
T
if and only if every member of S
is a member of T.
In this case we also say that
S
is a subset of
T
Venn diagram
To express formally the fact that
S
is a subset of
T,
we write
S T
Definition 4.3 (Formal counter part)

For any two sets
S
and
T,
S T: { for all x S: x S x T }.
Definition 4.4 (Strict subset)

For any sets
S
and
T,
S T if and only if
S T and there exists
x such that x T and x S
(which implies S not equal T).
If S T then S is a strict or proper
subset of T.
Venn diagram
Definition 4.5 (Empty set)

The empty set is a set with no members,
it is denoted by Ø.
Thus,
formally,
Ø is defined by the condition
that for any object x, x Ø.
Principle of extensionality

Let A and B any two sets.
If A B and B A then A = B.
Created by unroff & hptools.
© by HansPeter Bischof. All Rights Reserved (1998).
Last modified: 27/July/98 (12:14)