PlanetMath: properties of complement

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properties of complement

(Derivation)

Let be a set and are subsets of .

$(A^{\complement})^\complement=A$ .
Proof. $a\in (A^{\complement})^\complement$ iff $a\notin A^{\complement}$ iff $a\in A$ . $\qedsymbol$
$\emptyset^\complement = X$ .
Proof. $a\in \emptyset^\complement$ iff $a\notin \emptyset$ iff $a\in X$ . $\qedsymbol$
$X^\complement = \emptyset$ .
Proof. $a\in X^\complement$ iff $a\notin X$ iff $a\in \emptyset$ . $\qedsymbol$
$A\cup A^\complement = X$ .
Proof. $a\in A\cup A^\complement$ iff $a\in A$ or $a\in A^\complement$ iff $a\in A$ or $a\notin A$ iff $a\in X$ . $\qedsymbol$
$A\cap A^\complement =\emptyset$ .
Proof. $a\in A\cap A^\complement$ iff $a\in A$ and $a\in A^\complement$ iff $a\in A$ and $a\notin A$ iff $a\in \emptyset$ . $\qedsymbol$
$A\subseteq B$ iff $B^\complement\subseteq A^\complement$ .
Proof. Suppose $A\subseteq B$ . If $a\in B^\complement$ , then $a\notin B$ , so $a\notin A$ , or $a\in A^\complement$ . This shows that $B^\complement\subseteq A^\complement$ . On the other hand, if $B^\complement\subseteq A^\complement$ , then by applying what's just been proved, $A=(A^\complement)^\complement \subseteq (B^\complement)^\complement =B$ . $\qedsymbol$
$A\cap B=\emptyset$ iff $A\subseteq B^\complement$ .
Proof. Suppose $A\cap B=\emptyset$ . If $a\in A$ , then $a\in B^\complement$ , or $a\notin B$ , which implies that $A\cap B=\emptyset$ . Suppose next that $A\subseteq B^\complement$ . If there is $a\in A\cap B$ , then $a\in B$ and $a\in A$ . But the second containment implies that $a\in B^\complement$ , which contradicts the first containment. $\qedsymbol$
$A\setminus B = A\cap B^\complement$ , where the complement is taken in .
Proof. $a\in A\setminus B$ iff $a\in A$ and $a\notin B$ iff $a\in A$ and $a\in B^\complement$ iff $a\in A\cap B^\complement$ . $\qedsymbol$
(de Morgan's laws) $(A \cup B)^\complement = A^\complement \cap B^\complement$ and $(A \cap B)^\complement = A^\complement \cup B^\complement$ .
Proof. See here. $\qedsymbol$

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Cross-references: de Morgan's laws, complement, implies, iff, subsets
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This is version 2 of properties of complement, born on 2008-03-19, modified 2008-03-19.
Object id is 10419, canonical name is PropertiesOfComplement.
Accessed 130 times total.

Classification:

AMS MSC:

03E99 (Mathematical logic and foundations :: Set theory :: Miscellaneous)

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