Mathematics · Ch 1 — Mathematical Logic

Two switches in series

1.5.1

Two switches in series

Applying Logic to Switching Circuits

The algebra of logical statements has a striking twin in electrical switching networks — an analogy first pointed out by Claude Shannon in 1930. Just as a statement is either true or false, a switch in a circuit is either on or off, so the same connectives ( $\land$ , $\lor$ , $\sim$ ) that describe statements can describe circuits too.

A switch is a two-state device that controls whether current can flow through a branch of a circuit; switches are labelled $S, S_1, S_2, S_3, \dots$ . When a switch is closed (on), current passes through it; when it is open (off), it blocks the current. A lamp $L$ wired through a switch $S$ glows exactly when $S$ is closed and stays dark when $S$ is open.

Figure 1.1Fig. 1.1 — An open switch S, the two-state on/off device whose states mirror a statement's truth values

$Fig. 1.1 — Fig. 1.1 — An open switch S, the two-state on/off device whose states mirror a statement's truth values$

Drawn by us to help you understand the concept clearly, and verified to make sure it's accurate. For exams, practice from your textbook's own diagram.

What this figure shows. Figure 1.1 shows a single switch, denoted generically by $S$ (or $S_1, S_2, S_3, \dots$ when several appear in one circuit) — a two-state device that either allows or blocks current, exactly mirroring a statement's two possible truth values. Current flows left-to-right th …

Figure 1.2Fig. 1.2 — A single lamp L controlled by one switch S in a complete circuit

$Fig. 1.2 — Fig. 1.2 — A single lamp L controlled by one switch S in a complete circuit$

Drawn by us to help you understand the concept clearly, and verified to make sure it's accurate. For exams, practice from your textbook's own diagram.

What this figure shows. Figure 1.2 is the simplest complete circuit: one lamp $L$ wired in a loop with a battery through a single switch $S$ . Closing $S$ (turning it 'on') completes the circuit and the lamp glows; opening $S$ (turning it 'off') breaks the circuit and the lamp stays dark — the basic on/off ↔ true/false, glows ↔ s …

To connect this to logic, every switch in a circuit is matched to a statement letter. Two switches that are always in the same state (both open or both closed together) are called equivalent switches and share the same letter — for instance switch $S_1$ is written as $p$ . Two switches that are always in opposite states are complementary switches: if one switch is written $p$ , the other (its complement, $S'$ ) is written $\sim p$ . Note that the circuit diagram itself never shows which state a switch is actually in.

Because the real state of any switch is not fixed in advance, we list every possible combination of on/off values for all the switches involved and record the resulting lamp state for each combination — this record is called the Input-Output table (or switching table), the electrical counterpart of a truth table. By convention, $1$ stands for "on" (closed) and $0$ for "off" (open). …

Figure 1.3Fig. 1.3 — Two switches S₁ and S₂ connected in series with lamp L

$Fig. 1.3 — Fig. 1.3 — Two switches S₁ and S₂ connected in series with lamp L$

Drawn by us to help you understand the concept clearly, and verified to make sure it's accurate. For exams, practice from your textbook's own diagram.

What this figure shows. Figure 1.3 shows switches $S_1$ and $S_2$ wired one after another along the single path to the lamp $L$ , so current must pass through BOTH switches to reach the lamp. With $p$ for ' $S_1$ is on' and $q$ for ' $S_2$ is on', the lamp glows exactly when $p$ and $q$ are both true — the truth condition of the conjunction $p \land q$ …

Table 3Table 1.22 — Input-output table for two switches in series (p ∧ q)

p (S1)	q (S2)	p ∧ q
1	1	1
1	0	0

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