Design a logic network to control the lights on the staircase. Switch A is located at the top of the stairs, switch B is located halfway up the stairs and switch C is positioned at the bottom of the stairs. The main stairway in a block of flats has three switches for controlling the lights. 2.9ĭraw (i) the switch contact circuits and (ii) the AND/OR implementations for the following Boolean functions.į 2( A, B, C) = ( Ā + B)( B ¯ + C) + ( AB + C)į 3( A, B, C, D) = ( A + B + C)( Ā + D) + B C ¯ + A( B + D)( C ¯ + D) 2.11 2.4Ĭonstruct a truth table for the following functions and from the truth table obtain an expression for the inverse functions:į 3( A, B, C) = ( A + B ¯)(Amacr + B ¯ + C)įind the inverse of the following expressions and do not simplify your resultį 2( A, B, C, D) = A( B + C) + B D ¯( Ā + C)į 3( A, B, C, D, E) = 2.6Įxpand and simplify the following expressions using De Morgan's theorem.į 1( A, B, C) = ( A + B ¯ ) ( A B C ¯ ) ( A ¯ C ¯ )į 2( A, B, C) = ( A B + B ¯ C ) + ( B C ¯ + A B ¯ )į 3( A, B, C) = ( A B + B ¯ C ) + ( A C + A ¯ C ¯ ¯ ) 2.7īC + AD = ( B+ A)( B + A)( B + D)( A + C)( C + D) 2.8įor the following two 4-variable functionsį 2 = A + C + BD how many of the input minterms are included in each of these functions and how many are not? What are the minterm expressions for the two functions? Simplify both functions using the theorems of Boolean algebra. Prove that ( A + B)( Ā + C) = AC + ĀB without using perfect induction. Simplify each of the following expressions using the method of optional products:į 1( A, B, C, D) = A C ¯ + B C ¯ D + A B ¯ C + ACDį 2( A, B, C, D) = B + Ā B ¯ + ACD + A C ¯į 3( A, B, C, D) = B + ĀB D ¯ + AB C ¯ + A B ¯ D + A C ¯ D 2.3 Using the theorems of Boolean algebra simplify the following expressions:į 1 ( A, B, C, D) = B + BCD + B ¯ CD + AB + ĀB + B ¯ Cį 2( A, B, C, D) = ( AB+C + D)( C ¯ + D)( C ¯ + D)( C ¯ + D + E ¯)į 3( A, B, C) = B C ¯( C + A C ¯) + ( Ā + C ¯)( ĀB + ĀC) 2.2
Then we look at implementing and minimizing logic circuits. The first thing we shall examine in this chapter is what do we mean by an algebra and why are we able to skip between these various interpretations. We can use truth tables, borrowed from the theory of propositions, as given in Chapter 3, or we can use Venn diagrams, borrowed from set theory, as given in Chapter 1. In simplifying logic circuits, use is made of the different interpretations that can be put upon the operations and variables. This connection is not surprising as membership of a set, A, could be defined using a statements like ‘3 is a member of A’ which is either TRUE or FALSE.
You will probably have noticed that the operations of Λ (AND), ∨ (OR), and ¬ (NOT) used in Chapter 3 for propositions are very similar to the operations ∩ (AND), ∪ OR, and ′ (NOT) (complement) used for sets. This very simple algebra is very powerful as it forms the basis of computer hardware. The operation of OR (+) is then performed on two voltage inputs, using an OR gate, AND(.) using an AND gate and NOT is performed using a NOT gate. Usually, a high voltage represents TRUE (or 1), and a low voltage represents FALSE (or 0). It is particularly important because of its use in design of logic circuits. Mary Attenborough, in Mathematics for Electrical Engineering and Computing, 2003 4.1 Introductionīoolean algebra can be thought of as the study of the set with the operations + (or).
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