164 CHAPTER 3 Logic Equivalent Circuits Sometimes two circuits that look very different will actually have exactly the same conditions under which the light will be on. If we were to analyze the truth tables for the corresponding symbolic statements for such circuits, we would determine that they have identical answer columns. In other words, the corresponding symbolic statements are equivalent. Did You Know? Applications of Logic George Boole provided the key that would unlock the door to modern computing. Not until 1938, however, did Claude Shannon, in his master’s thesis at MIT, propose uniting the on–off capability of electrical switches with Boole’s two-value system of 0’s and 1’s. The operations AND, OR, and NOT and the rules of logic laid the foundation for computer gates. Such gates determine whether the current will pass. The closed switch (current flow) is represented as 1, and the open switch (no current flow) is represented as 0. For more information, see the Group Projects in MyLab Math. 0 0 AND Gate OR Gate NOT Gate 0 1 0 0 0 1 0 1 1 1 0 0 0 1 0 1 0 1 1 1 1 1 1 1 0 0 As you can see from the accompanying diagrams, the gates function in essentially the same way as a truth table. The gates shown here are the simplest ones, representing simple statements. There are other gates, such as the NAND and NOR gates, that are combinations of NOT, AND, and OR gates. The microprocessing unit of a computer uses thousands of these switches. Definition: Equivalent Circuits Equivalent circuits are two circuits that have equivalent corresponding symbolic statements. To determine whether two circuits are equivalent, we will analyze the answer columns of the truth tables of their corresponding symbolic statements. If the answer columns from their corresponding symbolic statements are identical, then the circuits are equivalent. Example 4 Are the Circuits Equivalent? Determine whether the two circuits are equivalent. q p r q p r p Solution The symbolic statement that represents the top circuit is p q r ( ). ∨ ∧ The symbolic statement that represents the bottom circuit is p q p r ( ) ( ). ∨ ∧ ∨ The truth tables for these statements are shown in Table 3.42. Now try Exercise 23 Table 3.42 p q r p ∨ q r ( ) ∧ p q ( ) ∨ ∧ p r ( ) ∨ T T T T T T T T T T T F T T F T T T T F T T T F T T T T F F T T F T T T F T T F T T T T T F T F F F F T F F F F T F F F F F T F F F F F F F F F Note that the answer columns for the two statements are identical. Therefore, p q r ( ) ∨ ∧ is equivalent to p q p r ( ) ( ) ∨ ∧ ∨ and the two circuits are equivalent. 7 Logic Gates Instructor Resources for Section 3.7 in MyLab Math • Objective-Level Videos 3.7 • Animation: Logic Gates • PowerPoint Lecture Slides 3.7 • MyLab Exercises and Assignments 3.7 • Chapter 3 Projects
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