2.5: Properties of Sets

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Let $A, B,$ and $C$ be sets and $U$ be the universal set. Then:

Commutative Law

Theorem $\PageIndex{1}$: Commutative Law

For all sets $A$ and $B$, $A \cup B =B \cup A $ and $A \cap B= B \cap A $

Proof

Let $x \in A \cup B$. Then $x \in A$ or $x \in B$. Which implies $x \in B$ or $x \in A$. Hence $x \in B \cup A$. Thus $A \cup B \subseteq B \cup A $. Similarly, we can show that $B \cup A \subseteq A \cup B $. Therefore, $A \cup B =B \cup A $.

Let $x \in A \cap B$. Then $x \in A$ and $x \in B$. Which implies $x \in B$ and $x \in A$. Hence $x \in B \cap A$. Thus $A \cap B \subseteq B \cap A $. Similarly, we can show that $B \cap A \subseteq A \cap B $. Therefore, $A \cap B =B \cap A $.

Distributive Law

Theorem $\PageIndex{2}$: Distributive Law

For all sets $A,B $ and $C$, $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$ and $A \cup (B \cap C) = (A \cup B) \cap (A \cup C).$

Proof

Let $ x \in A \cap (B \cup C) $.

Then $x \in A$ and $ x \in B \cup C$.

Thus $x \in A$ and $ x \in B $ or $x \in C$.

Which implies $x \in A$ and $ x \in B $ or $x \in A$ and $ x \in C $.

Hence $ x \in (A \cap B) \cup (A \cap C)$. Thus $A \cap (B \cup C) \subseteq (A \cap B) \cup (A \cap C)$. Similarly, we can show that $(A \cap B) \cup (A \cap C) \subseteq A \cap (B \cup C) $. Therefore, $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$.

We have illustrated using a Venn diagram:

$alt$

De Morgan's Laws

Theorem $\PageIndex{3}$: De Morgan's Law

$(A \cup B)^c = A^c \cap B^c$ and $(A \cap B)^c = A^c \cup B^c $

We have illustrated using a Venn diagram:

$alt$

Relative Complements

Theorem $\PageIndex{4}$: Relative Complements

$A \setminus (B \cup C) = (A \setminus B) \cap (A \setminus C)$ and $A \setminus (B \cap C) = (A \setminus B) \cup (A \setminus C).$

We have illustrated using a Venn diagram:

$alt$

Idempotents

Theorem $\PageIndex{5}$: Idempotents

$A \cap A=A$ and$A \cup A=A$.

Identity

Theorem $\PageIndex{6}$: Identity

$A \cap \emptyset= \emptyset$ and$A \cup \emptyset=A$.

Complements

Theorem $\PageIndex{7}$: Complements

$A \cap A^c= \emptyset$ and$A \cup A^c= U$.
$(A^c)^c=A$.
$\emptyset^c=U$.
$U^c=\emptyset$.

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Theorem \(\PageIndex{3}\): De Morgan's Law

Theorem \(\PageIndex{4}\): Relative Complements

Theorem \(\PageIndex{5}\): Idempotents

Theorem \(\PageIndex{6}\): Identity

Theorem \(\PageIndex{7}\): Complements