# 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:

### 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:

### 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:

## 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
1. $$A \cap A^c= \emptyset$$ and$$A \cup A^c= U$$.
2. $$(A^c)^c=A$$.
3. $$\emptyset^c=U$$.
4.  $$U^c=\emptyset$$.

This page titled 2.5: Properties of Sets is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by Pamini Thangarajah.