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subset (Definition)

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:

$\displaystyle x\in A\Rightarrow x\in B.$
The relation between $ A$ and $ B$ is then called set inclusion.

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\}$, then neither $ A \subseteq C$ (because $ t\in A$ but $ t\not\in C$) nor $ C \subseteq 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$. Similarly, we 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. The set $ A$ is called a proper subset of $ B$, if $ A\subset B$ and $ A\neq B$. In this case, we do not use $ A\subseteq B$.



"subset" is owned by Wkbj79. [ full author list (3) | owner history (2) ]
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See Also: empty set, superset, totally bounded, proof that all subgroups of a cyclic group are cyclic, property, uniqueness of cardinality

Also defines:  set inclusion
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Cross-references: proper subset, empty set, NOR, relation, implication
There are 443 references to this entry.

This is version 8 of subset, born on 2001-10-24, modified 2007-06-14.
Object id is 472, canonical name is Subset.
Accessed 19292 times total.

Classification:
AMS MSC: 03-00 (Mathematical logic and foundations :: General reference works )
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