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Understanding Multiplexers: Truth Tables and Boolean Expressions - Prof. Gary Locklair, Exams of Computer Science

Instructions on how to construct a multiplexer circuit using truth tables and boolean expressions. It explains the function of a multiplexer, how to create a truth table, and the steps to generate a boolean expression. The document also mentions the use of and, or, and not gates in creating the circuit.

Typology: Exams

Pre 2010

Uploaded on 07/22/2009

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23 October – Day 17
Wassn7 due
Chapter 13 – (goes together well with chapter 3)
Read first page and a half
note: talking about the CU of a CPU
A multiplexer is a device which accepts multiple inputs and produces a
single output. Simple example: an AND gate
Think of figure 13.1 as a “black box” which behaves as specified in the
truth table.
What should the black box do? (what’s it function?) Specify via truth
table.
Note that input “s” is a select input. If s is a 0, then the value of x0 is
passed to the output, f. If s is a 1, then the value of x1 is passed to the
output f.
Q: How do I construct the circuit which performs f (what’s inside the
black box?) Remember that the circuit is some combination of gates
(and, or, not, etc)
The Boolean function f can be defined by a Boolean expression
Q: How do I create a Boolean expression
To create the Boolean expression,
0.) Generate the truth table (given in example) – may have to test the
black box and determine outputs if not given!
1.) at each row where f = 1, create a product (and) of terms. That is,
for each term with a 1 value, use it directly; if the term has a 0 value,
use its complement (not)
2.) sum (or) all of the products together from step 1
Example 13.1
step 1.)
Row 3 = S’ X0 X1
Row 4 = S’ X0 X1
CSC 490 Course Notes and Outline, © Dr. Gary Locklair, Fall 2006
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23 October – Day 17 Wassn7 due Chapter 13 – (goes together well with chapter 3) Read first page and a half note: talking about the CU of a CPU A multiplexer is a device which accepts multiple inputs and produces a single output. Simple example: an AND gate Think of figure 13.1 as a “black box” which behaves as specified in the truth table. What should the black box do? (what’s it function?) Specify via truth table. Note that input “s” is a select input. If s is a 0, then the value of x0 is passed to the output, f. If s is a 1, then the value of x1 is passed to the output f. Q: How do I construct the circuit which performs f (what’s inside the black box?) Remember that the circuit is some combination of gates (and, or, not, etc) The Boolean function f can be defined by a Boolean expression Q: How do I create a Boolean expression To create the Boolean expression, 0.) Generate the truth table (given in example) – may have to test the black box and determine outputs if not given! 1.) at each row where f = 1, create a product (and) of terms. That is, for each term with a 1 value, use it directly; if the term has a 0 value, use its complement (not) 2.) sum (or) all of the products together from step 1 Example 13. step 1.) Row 3 = S’  X 0  X 1 ’ Row 4 = S’  X 0  X 1 CSC 490 Course Notes and Outline, © Dr. Gary Locklair, Fall 2006

Row 6 = S  X 0 ‘  X 1 Row 8 = S  X 0  X 1 step 2.) now, ( ) + ( ) + ( ) + ( ) write out whole expression note: “logically” we want a “1” answer for the row, so if we sum (and), then we need to ensure that each term is a “1;” therefore, if it is a “0,” we need to complement (not) it This is the “Disjunction Normal Form” – or – “Sum of Products Form” Now it is possible to create the circuit directly by inspection (use and, or, not which is a complete set) Note circuit in Fig 13.3 The final gate is an OR gate with 4 inputs – this is from step 2.) above since there are 4 products (rows in the truth table) with a 1 output. Note that the inputs to the OR gate are 4 AND gates. These represent the products of the 3 terms. WASSN 8 – problem 4, page 89. How? Create a truth table. let students “struggle” with truth table on their own (answer questions next time) x0 s | y0 y 0 0 0 - 0 1 - 0 1 0 1 - 1 1 - 1 output is a circuit of and, or, not Systems Assignment # next part of “Giant Brains” video from “Machine that changed the world” series CSC 490 Course Notes and Outline, © Dr. Gary Locklair, Fall 2006