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Inclusion

Definition 3.2.1   Let $ A$ and $ B$ be two sets. The set $ A$ is called a subset of $ B$ if the following property holds:

$\displaystyle \forall x, \; x \in A \Longleftrightarrow x \in B.$    

We denote this by $ A \subset B$.

Figure 2: Inclusion (subset).
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\mbox{\epsfig{file=inclusion.eps,height=4cm}}
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Example 3.2.2   Let $ A=\{ a,b,\%, \& \}$ and $ B=\{ 1,a,b,\char93 ,\%,\&,0 \}$. Then $ A \subset B$.

If $ A$ is not a subset of $ B$, we denote $ A \not\subset B$. For example, the set $ \{ a,b \}$ is not a subset of the set $ \{ a, c,d
\}$.

Proposition 3.2.3   If $ A$, $ B$ and $ C$ are three sets, such that $ A \subset B$ and $ B \subset C$, then $ A \subset C$.

Here too, we can prove the proposition either with truth tables (v.s. 3.14). Now we fix some set $ E$, sometimes called an universal set. We write $ \mathcal{P}(E)$ for the set of all subsets of $ S$, called the power set of $ E$.

Example 3.2.4   Let $ E=\{ a,b,c \}$. Then

$\displaystyle \mathcal{P}(E)= \{ \empty, \{ a \},\{ b \}, \{ c \},\{ a,b \},\{ a,c \},
 \{ b,c \},\{ a,b,c \} \}.$    

Example 3.2.5   Consider the empty set $ \emptyset$. We have:


next up previous contents
Next: Equality Up: Operations with sets Previous: Operations with sets   Contents
root 2002-06-10