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'subset'
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| Title of object: |
subset |
| Canonical Name: |
Subset |
| Type: |
Definition |
| Created on: |
2001-10-24 12:47:54 |
| Modified on: |
2003-03-26 03:46:27 |
| Classification: |
msc:03-00 |
Revision comment (for changes between this and next version):
| Changes for correction #2088 ('proper subset'). |
Preamble:
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{graphicx}
\usepackage{xypic} |
Content:
Given two sets $A$ and $B$, we say that $A$ is a subset of $B$ (which we denote as $A\subseteq B$ or simply $A\subset B$) if every element of $A$ is also in $B$. That is, the following implication holds:
$$x\in A\Rightarrow x\in B.$$
Some examples:
The set $A=\{d,r,i,t,o\}$ is a subset of the set $B=\{p,e,d,r,i,t,o\}$ because every element of $A$ is also in $B$. That is, $A\subseteq B$.
On the other hand, if $C=\{p,e,d,r,o\}$ neither $A$ is a subset of $C$ (because $t\in A$ but $t\not\in C$) nor $C$ is a subset of $A$ (because $p\in C$ but $p\not\in A$).
The fact that $A$ is not a subset of $C$ is written as $A\not\subseteq C$. And then, in this example we also have $C\not\subseteq A$.
If $X\subseteq Y$ and $Y\subseteq X$, it must be the case that $X=Y$.
Every set is a subset of itself, and the empty set is a subset of every other set. |
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