Simplification of compound statements becomes especially concrete in switching networks. A switch may be open or closed, just as a statement may be false or true. Logical equivalence then says when two networks behave the same under every switch configuration. In this lesson, switching networks and logical equivalence will connect series and parallel circuits with conjunction and disjunction. :::definition[Switching Network] A switching network is a network made of wires and switches connecting two terminals $T_1$ and $T_2$. The purpose of the network is to determine whether current can flow from $T_1$ to $T_2$. ::: :::definition[Open Switch] A switch is open if no current flows through it. An open switch is represented by $0$. ::: :::definition[Closed Switch] A switch is closed if current flows through it. A closed switch is represented by $1$. ::: :::definition[Single Switch Network] A network containing one switch $p$ is represented by $p$. Current flows through the network exactly when $p$ is closed. ::: :::definition[Parallel Network] A network is a parallel network when current flows if at least one of the switches is closed. For switches $p$ and $q$, the parallel network is represented by $p\lor q$. ::: :::definition[Series Network] A network is a series network when current flows only if all switches in the path are closed. For switches $p$ and $q$, the series network is represented by $p\land q$. ::: Use the flow tester to connect switching networks with truth values. Set each switch to open or closed, then choose a network expression. A series connection behaves like and because every switch in the path must be closed. A parallel connection behaves like or because current can pass through at least one closed path. :::scientific-preview[Switching Network Flow Tester] ::: :::example[Networks Represented by Logical Statements] A network with one switch $p$ is represented by $p$. A network with two switches $p$ and $q$ in parallel is represented by $p\lor q$. A network with two switches $p$ and $q$ in series is represented by $p\land q$. Thus, disjunction corresponds to parallel connection, and conjunction corresponds to series connection. ::: :::theorem[Logical Equivalence of Switching Networks] Let $N_1$ and $N_2$ be switching networks represented by statements $s_1$ and $s_2$. If $s_1\Leftrightarrow s_2$, then the two networks allow current to flow under exactly the same switch conditions. ::: :::proof Given that $N_1$ and $N_2$ are switching networks represented by $s_1$ and $s_2$, and $s_1\Leftrightarrow s_2$. To prove that $N_1$ and $N_2$ behave in the same way. Since $s_1$ and $s_2$ have the same truth value for every assignment of truth values to their switches, and since truth value $1$ means current flows while truth value $0$ means current does not flow, the networks conduct under exactly the same configurations. Hence, the two networks are equivalent. $\square$ ::: :::example[Coupled Switching Network] Consider a switching network represented by $$\begin{equation*} (p\lor q\lor r)\land(p\lor t\lor\neg q)\land(p\lor\neg t\lor r). \end{equation*}$$ Here some switches are coupled, meaning that one switch may depend on another. For example, switches labeled $t$ and $\neg t$ are not independent. ::: :::solved-problem Simplify the network represented by $(p\lor q\lor r)\land(p\lor t\lor\neg q)\land(p\lor\neg t\lor r)$. ::: :::solution Given that the network statement is $(p\lor q\lor r)\land(p\lor t\lor\neg q)\land(p\lor\neg t\lor r)$. Use the distributive pattern $(p\lor A)\land(p\lor B)\Leftrightarrow p\lor(A\land B)$. Then $$\begin{equation*} (p\lor q\lor r)\land(p\lor t\lor\neg q)\land(p\lor\neg t\lor r)\Leftrightarrow p\lor[(q\lor r)\land(t\lor\neg q)\land(\neg t\lor r)]. \end{equation*}$$ The bracket simplifies to $r\land(t\lor\neg q)$ by distribution and inverse laws. Therefore, $$\begin{equation*} (p\lor q\lor r)\land(p\lor t\lor\neg q)\land(p\lor\neg t\lor r)\Leftrightarrow p\lor[r\land(t\lor\neg q)]. \end{equation*}$$ Hence, the original network can be replaced by a simpler equivalent network. ::: :::corollary If a switching network is represented by a compound statement $s$, and $s\Leftrightarrow t$, where $t$ has fewer switch variables or fewer repeated switch occurrences, then the network represented by $t$ is an equivalent simplified network. ::: :::proof Given that $s\Leftrightarrow t$. To prove that the network represented by $t$ is equivalent to the network represented by $s$. Since $s$ and $t$ are logically equivalent, they have the same truth value under every switch configuration. Thus, the two networks either both allow current to flow or both block current under the same conditions. Hence, the simplified network may replace the original network. $\square$ ::: :::solved-problem Represent a switching network with $p$ and $q$ in series, placed in parallel with switch $r$. ::: :::solution Let $p$ and $q$ be in series. The series part is $p\land q$. This series part is placed in parallel with $r$, so the whole network is $$\begin{equation*} (p\land q)\lor r. \end{equation*}$$ Hence, current flows when both $p$ and $q$ are closed, or when $r$ is closed. ::: :::exercise Write logical expressions for: (1) switches $p$ and $q$ in parallel; (2) switches $p$ and $q$ in series; (3) switch $p$ in series with a parallel pair $q,r$; (4) switch $p$ in parallel with a series pair $q,r$. ::: :::answer (1) $p\lor q$. (2) $p\land q$. (3) $p\land(q\lor r)$. (4) $p\lor(q\land r)$. ::: :::faq[Frequently Asked Questions] Q: What does $1$ mean in a switching network? A: It means the switch is closed and current can pass through that switch. Q: Which connective represents parallel connection? A: Parallel connection is represented by $\lor$. Q: Which connective represents series connection? A: Series connection is represented by $\land$. ::: :::call-to-action[Connect Networks and Logic] subtitle: Practise translating series and parallel networks before studying NAND and NOR connectives. button: Next Lesson | /discrete-mathematics/mathematical-logic/logical-equivalence-and-the-laws-of-logic/nand-and-nor-connectives button-ghost: Previous Lesson | /discrete-mathematics/mathematical-logic/logical-equivalence-and-the-laws-of-logic/simplification-of-compound-statements :::