
# 2.3: Properties of Sets

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

### Commutative Law

1. $$A \cup B =B \cup A$$ and

$$A \cap B= B \cap A$$

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

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

We have illustrated using a Venn diagram:

Example $$\PageIndex{1}$$:

Consider $$A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$$:

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, $$A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$$.

### De Morgan's Laws

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

We have illustrated using a Venn diagram:

Example $$\PageIndex{2}$$:

Consider $$(A \cup B)^c = A^c \cap B^c$$:

Theorem $$\PageIndex{3}$$

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

Proof

Exercise.

### Relative Complements

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

We have illustrated using a Venn diagram:

Example $$\PageIndex{3}$$:

Consider $$A \setminus (B \cup C) = (A \setminus B) \cap (A \setminus C)$$:

Proof:

Discussed in class.

## Idempotents

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

## Identity

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

## Complements

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

8. $$(A^c)^C=A$$.

9. $$\emptyset^c=U$$.

10. $$U^c=\emptyset$$.