Algebra of Sets

Overview

The study of the operations of union ($\cup$), intersection ($\cap$), and set difference ($-$), together with the inclusion relation ($\subseteq$), goes by the algebra of sets.

Symmetric Difference

Define the symmetric difference of sets $A$ and $B$ as $A△B=(A-B)\cup (B-A)$

Cartesian Product

Given two sets $A$ and $B$, the Cartesian product $A\times B$ is defined as: $A\times B=\{⟨x,y⟩\mid x\in A\wedge y\in B\}$

We can also form (something like) the Cartesian product of infinitely many sets, provided that the sets are suitably indexed. Let $I$ be an index set and $H$ a function whose domain includes $I$. Define ${\text{bigtimes}}_{i\in I}H(i)=\{f\mid f\text{ is a function with domain }I\text{ and }\forall i\in I,f(i)\in H(i)\}$

Laws

Commutative Laws

For any sets $A$ and $B$, \begin{align*} A \cup B & = B \cup A \\ A \cap B & = B \cap A \end{align*}

Associative Laws

For any sets $A$ and $B$, \begin{align*} A \cup (B \cup C) & = (A \cup B) \cup C \\ A \cap (B \cap C) & = (A \cap B) \cap C \end{align*}

Distributive Laws

For any sets $A$, $B$, and $C$, \begin{align*} A \cap (B \cup C) & = (A \cap B) \cup (A \cap C) \\ A \cup (B \cap C) & = (A \cup B) \cap (A \cup C) \end{align*}

More generally, for any sets $A$ and $\mathcal{B}$, \begin{align*} A \cup \bigcap \mathscr{B} & = \bigcap\, \{A \cup X \mid X \in \mathscr{B}\}, \text{ for } \mathscr{B} \neq \varnothing \\ A \cap \bigcup \mathscr{B} & = \bigcup\, \{A \cap X \mid X \in \mathscr{B}\} \end{align*}

For any sets $A$, $B$, and $C$, \begin{align*} A \times (B \cap C) & = (A \times B) \cap (A \times C) \\ A \times (B \cup C) & = (A \times B) \cup (A \times C) \\ A \times (B - C) & = (A \times B) - (A \times C) \end{align*}

In addition, \begin{align*} A \times \bigcup \mathscr{B} & = \bigcup\, \{A \times X \mid X \in \mathscr{B}\} \\ A \times \bigcap \mathscr{B} & = \bigcap\, \{A \times X \mid X \in \mathscr{B}\} \end{align*}

De Morgan’s Laws

For any sets $A$, $B$, and $C$, \begin{align*} C - (A \cup B) & = (C - A) \cap (C - B) \\ C - (A \cap B) & = (C - A) \cup (C - B) \end{align*}

More generally, for any sets $C$ and $\mathcal{A}\ne \varnothing$, \begin{align*} C - \bigcup \mathscr{A} & = \bigcap\, \{C - X \mid X \in \mathscr{A}\} \\ C - \bigcap \mathscr{A} & = \bigcup\, \{C - X \mid X \in \mathscr{A}\} \end{align*}

Monotonicity

Let $A$, $B$, and $C$ be arbitrary sets. Then

• $A\subseteq B\Rightarrow A\cup C\subseteq B\cup C$,
• $A\subseteq B\Rightarrow A\cap C\subseteq B\cap C$,
• $A\subseteq B\Rightarrow \bigcup A\subseteq \bigcup B$

In addition,

• $A\subseteq B\Rightarrow A\times C\subseteq B\times C$

Antimonotonicity

Let $A$, $B$, and $C$ be arbitrary sets. Then

• $A\subseteq B\Rightarrow C-B\subseteq C-A$,
• $\varnothing \ne A\subseteq B\Rightarrow \bigcap B\subseteq \bigcap A$

Cancellation Laws

Let $A$, $B$, and $C$ be sets. If $A\ne \varnothing$,

• $(A\times B=A\times C)\Rightarrow B=C$
• $(B\times A=C\times A)\Rightarrow B=C$

Index Sets

Let $I$ be a set, called the index set. Let $F$ be a function whose domain includes $I$. Then we define ${\bigcup }_{i\in I}F(i)=\bigcup \{F(i)\mid i\in I\}$ and, if $I\ne \varnothing$, ${\bigcap }_{i\in I}F(i)=\bigcap \{F(i)\mid i\in I\}$

Function Sets

For sets $A$ and $B$, the collection of functions $F$ from $A$ into $B$ is: ${}^{A}B=\{F\mid F:A\to B\}$ ${}^{A}B$ is read as “$B$-pre-$A$”. It is often written as $B^A$ instead.

Bibliography

• Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).