For any set $S$, $(\mathcal{P}(S), \cap, \cup, \overline{\phantom{A}})$ forms a Boolean algebra:

• $A \cap B$ (intersection) is the meet
• $A \cup B$ (union) is the join
• $\overline{A} = S \setminus A$ (complement) is negation
• $\emptyset$ is 0, $S$ is 1

All axioms are satisfied. This is the prototypical Boolean algebra, and Stone's representation theorem shows every Boolean algebra is isomorphic to a field of sets.

Digital circuits implement Boolean functions using logic gates:

Basic gates:

• NOT gate ($\neg$): Inverter
• AND gate ($\land$): Output 1 if all inputs are 1
• OR gate ($\lor$): Output 1 if any input is 1
• XOR gate ($\oplus$): Output 1 if odd number of inputs are 1

Universal gates:

• NAND gate: NOT-AND, $\neg(a \land b)$. Universal: any Boolean function can be built using only NAND gates.
• NOR gate: NOT-OR, $\neg(a \lor b)$. Also universal.

For instance, NOT can be built from NAND: $\neg a = a \text{ NAND } a$. AND from NAND: $a \land b = \neg(a \text{ NAND } b)$.

In electrical engineering, Boolean algebra models switching circuits:

• Closed switch (conducts): 1
• Open switch (blocks): 0
• Series connection: AND
• Parallel connection: OR

A circuit with switches $a, b, c$ in configuration $(a \land b) \lor c$ conducts when ($a$ and $b$ are both closed) or ($c$ is closed).

This application, pioneered by Shannon, enabled systematic circuit design replacing trial-and-error with algebraic manipulation.

For a positive integer $n$, the divisors of $n$ with divisibility order form a lattice:

• $a \land b = \gcd(a,b)$ (meet)
• $a \lor b = \text{lcm}(a,b)$ (join)
• Minimum: 1, Maximum: $n$

For $n = 30$, divisors $\{1, 2, 3, 5, 6, 10, 15, 30\}$ form a lattice. This is a Boolean algebra only when $n$ is square-free (no repeated prime factors). For $n = p_1 p_2 \cdots p_k$ (distinct primes), the divisor lattice is isomorphic to $\mathcal{P}(\{p_1, \ldots, p_k\})$.

States of a deterministic finite automaton (DFA) can be minimized using lattice operations. The partition lattice of states, ordered by refinement, guides the minimization algorithm. Equivalent states are merged using fixed-point iteration on this lattice structure.