3.7 Switching Circuits 161 Table 3.38 p q Light p q ∧ T T on (T) T T F off (F) F F T off (F) F F F off (F) F Table 3.39 p q Light p q ∨ T T on (T) T T F on (T) T F T on (T) T F F off (F) F CASE 1: Both switches are closed; that is, p is T and q is T. The light is on, T. CASE 2: Switch p is closed and switch q is open; that is, p is T and q is F. The light is off, F. CASE 3: Switch p is open and switch q is closed; that is, p is F and q is T. The light is off, F. CASE 4: Both switches are open; that is, p is F and q is F. The light is off, F. Table 3.38 summarizes the results. The on–off results are the same as the truth table for the conjunction p q ∧ if we think of “on” as true and “off” as false. Series Circuits Switches in series will always be represented with a conjunction, .∧ Another type of electric circuit used in the home is the parallel circuit, in which there are two or more paths that the current can take. If the current can pass through either path or both (see Fig. 3.15), the light will go on. The letter T will be used to represent both a closed switch and the bulb lighting. The letter F will represent an open switch and the bulb not lighting. Thus, we have four possible cases as shown in Table 3.39. q p Figure 3.15 CASE 1: Both switches are closed; that is, p is T and q is T. The light is on, T. CASE 2: Switch p is closed and switch q is open; that is, p is T and q is F. The light is on, T. CASE 3: Switch p is open and switch q is closed; that is, p is F and q is T. The light is on, T. CASE 4: Both switches are open; that is, p is F and q is F. The light is off, F. Table 3.39 summarizes the results. The on–off results are the same as the p q ∨ truth table if we think of “on” as true and “off” as false. Parallel Circuits Switches in parallel will always be represented with a disjunction, .∨ Sometimes it is necessary to have two or more switches in the same circuit that will both be open at the same time and both be closed at the same time. In such circuits, we will use the same letter to represent both switches. For example, in the circuit shown in Fig. 3.16, there are two switches labeled p. Therefore, both of these switches must be open at the same time and both must be closed at the same time. One of the p switches cannot be open at the same time the other p switch is closed. We can now combine some of these basic concepts to analyze more circuits. q p p Figure 3.16
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