Application: Switching Circuits

The algebra of statements turns out to describe electrical switching circuits exactly, which is why logic underlies the design of every digital device. This application closes the chapter and is a regular source of examination problems.

The dictionary. A switch in a circuit is represented by a statement (a letter $p, q, \ldots$). A switch is in one of two states, matched to the two truth values:

• closed — current passes — corresponds to $T$;
• open — current is blocked — corresponds to $F$.

Two switches $p$ and $q$ can be wired in two basic ways:

• Series (one after the other): current flows only if both are closed. A series connection is the conjunction $p \wedge q$.
• Parallel (side by side, on separate branches): current flows if at least one is closed. A parallel connection is the disjunction $p \vee q$.

Two switches that always act oppositely — one closed exactly when the other is open — are represented by $p$ and $\sim p$ and are called complementary switches. Switches wired to open and close together are written with the same letter.

Symbolic form of a circuit. Read the circuit from one terminal to the other, writing $\wedge$ for each series link and $\vee$ for each parallel split, respecting the grouping of the branches with brackets. For instance, a switch $p$ in series with a parallel pair $q, r$ has the symbolic form $p \wedge (q \vee r)$.

Simplifying a circuit. A circuit "works" (lets current through) exactly when its symbolic form is true. Two circuits with logically equivalent symbolic forms therefore behave identically, so a complicated circuit can be replaced by any simpler equivalent one — using fewer switches, which is cheaper to build. The method is:

1. write the circuit's symbolic form;
2. simplify it with the algebra of statements (Section 5), aiming for complement pairs ($p \wedge \sim p \equiv \mathbf{c}$, $p \vee \sim p \equiv \mathbf{t}$) and absorption;
3. read the simplified form back as a circuit.

For example, the form $(p \wedge q) \vee (p \wedge \sim q)$ simplifies, by the distributive law and then the complement and identity laws, to $p \wedge (q \vee \sim q) \equiv p \wedge \mathbf{t} \equiv p$ — a two-branch, four-switch circuit that does exactly the same job as a single switch $p$. This is precisely the kind of saving the algebra of statements delivers in practice.

The two circuits in that example, described in words between the same pair of terminals L and R, are: …