162 CHAPTER 3 Logic p q p Figure 3.17 Example 1 Representing a Switching Circuit with Symbolic Statements a) Write a symbolic statement that represents the circuit shown in Fig. 3.16. b) Construct a truth table to determine when the light will be on. Solution a) In Fig. 3.16, there is a blue switch p on the left, and to the right there is a branch containing a red switch p and a switch q. The current must flow through the blue switch p into the branch on its right. Therefore, the blue switch p is in series with the branch containing the red switch p and switch q. After the current flows through the blue switch p it reaches the branch on its right. At this point, the current has the option of flowing into the red switch p, switch q, or both of these switches. Therefore, the branch containing the red switch p and switch q is a parallel branch. We say these two switches are in parallel. The entire circuit in symbolic form is p p q ( ). ∧ ∨ Note that the parentheses are very important. Without parentheses, the symbolic statement could be interpreted as p p q ( ) . ∧ ∨ The diagram for p p q ( ) ∧ ∨ is illustrated in Fig. 3.17. We shall see later in this section that the circuits in Fig. 3.16 and Fig. 3.17 are not the same. Now try Exercise 5 b) The truth table for the statement (Table 3.40) indicates that the light will be on only in the cases in which p is true or when switch p is closed. 7 Table 3.40 p q p ∧ p q ( ) ∨ T T T T T T F T T T F T F F T F F F F F 1 3 2 Learning Catalytics Keyword: Angel-SOM-3.7 (See Preface for additional details.) Example 2 Representing a Switching Circuit with Symbolic Statements a) Write a symbolic statement that represents the circuit in Fig. 3.18. b) Construct a truth table to determine when the light will be on. r p p q Figure 3.18 Solution a) The upper branch of the circuit contains two switches p and q in series. We represent this branch with the statement p q. ∧ The lower branch of the circuit contains two switches p and r in parallel. We represent this branch with the statement p r. ∨ The upper branch is in parallel with the lower branch. Putting the two branches together, we get the statement p q p r ( ) ( ). ∧ ∨ ∨ b) The truth table for the statement (Table 3.41) shows that cases (rows) 6 and 8 are false. Thus, the light will be off in these two cases and on in all other cases. Now try Exercise 9 Table 3.41 p q r p q ( ) ∧ ∨ p r ( ) ∨ T T T T T T T T F T T T T F T F T T T F F F T T F T T F T T F T F F F F F F T F T T F F F F F F 1 3 2
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