Electrical Network

1. Current flows between $T_1$ and $T_2$ when switch $P$ is closed i.e. $P$ is true OR
2. The current flows between $T_1$ and $T_2$ when switches $Q$ and $R$ are closed i.e. $Q \wedge R$ is true. The required compound statement is $P \vee (Q \wedge R)$
3. To find possible switch settings, draw a truth table $P \vee (Q \wedge R)$

P	Q	R	Q ∧ R	P ∨ (Q ∧ R)	Current flows
T	T	T	T	T	Yes
T	T	F	F	T	Yes
T	F	T	F	T	Yes
T	F	F	F	T	Yes
F	T	T	T	T	Yes
F	T	F	F	F	No
F	F	T	F	F	No
F	F	F	F	F	No

Possible switch settings

P	Q	R
Closed	Closed	Closed
Closed	Closed	Closed
Closed	Closed	Closed
Closed	Closed	Closed
Closed	Closed	Closed

1. Construct compound statements that correspond to the networks

The current will flow when all three switches $P$, $Q$, and $R$ are closed i.e. $P \wedge Q \wedge R$

The required compound statement is $P \wedge Q \wedge R$

The required compound statement is $P \vee Q$

The required compound statement is $(P \wedge Q) \vee (R \wedge S)$

The required compound statement is $P \vee Q \vee R$

The required compound statement is $P \wedge (Q \vee (R \wedge S))$

The required compound statement is $(P \vee Q \vee R) \wedge S$

2. In electrical network of (ii) find possible switch settings that will allow the current to flow between $T_1$ and $T_2$

(ii) $(P \vee Q) \wedge R$

P	Q	R	P ∨ Q	(P ∨ Q) ∧ R
T	T	T	T	T
T	T	F	T	F
T	F	T	T	T
T	F	F	T	F
F	T	T	T	T
F	T	F	T	F
F	F	T	F	F
F	F	F	F	F

Possible switch settings

P	Q	R
Closed	Closed	Closed
Closed	Open	Closed

From statements to network

Example

Draw a network for the statement $(P \vee Q) \wedge (R \wedge S)$

Corresponding network is shown below

Draw networks for the following statements

2. $[(P \wedge Q) \wedge (R \vee S)]$
3. $(Q \vee (R \vee S) \vee P)$

These operate as follows

1. When one switch is closed, the other one closes also
2. When one switch is closed, the other one opens

Refer to the diagram

The compound relating to flow of electrical current is given

$(P \wedge Q) \vee [P \wedge (\sim Q \vee R)]$

To find possible switch settings that will allow the current to flow between $T_1$ and $T_2$

– Draw a truth table for $(P \wedge Q) \vee [P \wedge (\sim Q \vee R)]$

P	Q	R	P ∧ Q	~Q	~Q ∨ R	P ∧ (~Q ∨ R)	(P ∧ Q) ∨ [P ∧ (~Q ∨ R)]
T	T	T	T	F	T	T	T
T	T	F	T	F	F	F	T
T	F	T	F	T	T	T	T
T	F	F	F	T	T	T	T
F	T	T	F	T	F	F	F
F	T	F	F	F	F	F	F
F	F	T	F	T	T	F	F
F	F	F	F	T	T	F	F

Possible switch settings

P	Q	R
Closed	Closed	Closed
Closed	Closed	Open
Closed	Open	Closed
Closed	Open	Open

Example

Without using a truth table draw a sample network for the statement

$(P \wedge Q) \vee [P \wedge (\sim Q \vee R)]$

$(P \wedge Q) \vee [P \wedge (\sim Q \vee R)] = P \wedge (Q \vee (\sim Q \vee R))$ ….. distributive

$= P \wedge ((Q \vee \sim Q) \vee R)$ ……. associative

$= P \wedge (T \vee R)$ ….. Complement

$= P \wedge T$ ….. Identity

$= P$ ….. Identity

The statement simplifies to $P$

The corresponding network is as follows

For a statement which on simplifying ends up at $F$, network drawn is as follows

For a statement which upon simplifying yields $T$, network is drawn as follows

1. For each of the networks shown below, find a compound statement that represents it

2. (a) Draw network for the corresponding statement

i) $(P \wedge \sim Q) \vee (Q \wedge P)$ ii) $(P \wedge \sim Q) \vee (Q \wedge \sim R)$ iii) $P \rightarrow Q \equiv \sim P \vee Q$

iv) $(P \rightarrow Q) \wedge (P \vee Q) \equiv (\sim P \vee Q) \wedge (P \vee Q)$

(b) Simplify the statement in 2 (iv) using the laws of algebra of propositions and draw a simple network

More examples

i) Write down compound statements for the following networks

2. For each of these sentences draw a simple network

a) $P \wedge (\sim Q \rightarrow \sim P)$

b) $\sim (P \vee Q) \rightarrow R$

c) $P \wedge \sim P$

3. Given a truth table

P	Q	R	Output
T	T	T	F
T	T	F	T
T	F	T	T
T	F	F	F
F	T	T	T
F	T	F	F
F	F	T	F
F	F	F	T

a) Construct a statement having this truth table

b) Draw the electrical network