2.5: Properties of Sets

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May 20, 2022
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- 2.3: Venn Diagrams and Euler Diagrams
- 2.E: Basic Concepts of Sets (Exercises)

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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

Commutative Law

Distributive Law

De Morgan's Laws

Theorem 2.5.3\PageIndex{3}: De Morgan's Law

Relative Complements

Theorem 2.5.4\PageIndex{4}: Relative Complements

Idempotents

Theorem 2.5.5\PageIndex{5}: Idempotents

Identity

Theorem 2.5.6\PageIndex{6}: Identity

Complements

Theorem 2.5.7\PageIndex{7}: Complements

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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