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4.6.  Subset

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

[larr] Venn diagram

To express formally the fact that S is a subset of T, we write S [sube] T

Definition 4.3 (Formal counter part)

For any two sets S and T, S [sube] T: { for all x [isin] S: x [isin] S [larr] x [isin] T }.

Definition 4.4 (Strict subset)

For any sets S and T, S [subset] T if and only if S [sube] T and there exists x such that x [isin] T and x [notin] S (which implies S not equal T).
If S [subset] T then S is a strict or proper subset of T.

[larr] 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 [notin] Ø.

Principle of extensionality

Let A and B any two sets. If A [sube] B and B [sube] A then A = B.

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Last modified: 27/July/98 (12:14)