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Chapter 2 Switching Algebra and Logic Circuits

7400 Datasheet

Logic Gates Circuit Symbols Truth Table Boolean Expression Daisy Chain Multiple Inputs

Figure 2.1 Symbols for OR and AND gates. Figure 2.3 A NOT gate.

Figure 2.2 AND gate implementation of Property 2b.

OR (written as +) 1 a + b (read a OR b) is 1 if and only if a = 1 or b = 1 or both AND (written as  or simply two variables concatenated) a  b = ab (read a AND b) is 1 if and only if a = 1 and b = 1. NOT (written) a (read NOT a) is 1 if and only if a = 0

A water sprinkler should turn on between 4am and 5am in the morning only when a rain sensor indicates that it has not rained in the last 48 hours. T = 1 if the time is between 4am and 5am R = 1 if it has rained in the last 48 hours Create a truth table and design a circuit that would perform this task.

The Book Inside the cover…

Boolean Theorems aa = ? a1 = ? a0 = ? aa’ = ? a’’ = ? a(b + c) = ?

Boolean Theorems a + a = ? a + 1 = ? a + 0 = ? a + a’ = ?

Proof Proof of theorem P8b.

De Morgan’s Theorems (AB)’ = ? (A + B) = ? Exclusive Or A + B = ? (A + B)’ = ?

DEFINITIONS Example: f = a’b’c + acd + a’b’c’d A literal is the appearance of a variable or its complement. A product term is one or more literals connected by AND operators. A standard product term, also minterm is a product term that includes each variable of the problem, either uncomplemented or complemented. A sum of products expression (often abbreviated SOP) is one or more product terms connected by OR operators.

A minimum sum of products expression is one of those SOP expressions for a function that has the fewest number of product terms. If there is more than one expression with the fewest number of terms, then minimum is defined as one or more of those expressions with the fewest number of literals. (1)xyz + xyz + xyz + xyz + xyz5 terms, 15 literals (2)xy + xy + xyz3 terms, 7 literals (3) xy + xy + xz3 terms, 6 literals (4) xy + xy + yz3 terms, 6 literals

DEFINITIONS Example: f = (a’+b’+c)(a+c+d)(a’+b’+c’+d) A sum term is one or more literals connected by OR operators. A standard sum term, also called a maxterm, is a sum term that includes each variable of the problem, either uncomplemented or complemented. A product of sums expression (POS) is one or more sum terms connected by AND operators.

Announcements The 12:30pm lab should first go to the Linc Lab on the 1 st floor of the library on 2/3/2015. The 4:00pm lab should first go to the Info Lab 1 on the 2 nd floor of the library on 2/3/2015. After you pick up your tablet from the library at the times above, then you go to room 315 of the science building to begin lab.

Announcements Work in groups of 2 in lab. All 27 students in the first lab section can begin lab at 12:30pm. Bring your book and lab handout to lab on Tuesday. The due date for Homework 2 has been moved to Thursday, February, 5 th.

Chapter 2 Problems Homework 2 (Page 100) Problems 2(f), 3(a), etc.

Figure 2.5 Minimum sum of product implementation of f. Figure 2.6 Circuit with only uncomplemented inputs.

Digital Logic Design Construct a circuit that will add two 2- digit binary numbers together (A 1 A 0 and B 1 B 0 ) and display the result as a binary number (Z 3 Z 2 Z 1 Z 0 ). The input numbers A 1 A 0 and B 1 B 0 can be handled with four data switches and the output Z 3 Z 2 Z 1 Z 0 can be displayed using four LED’s.

f is 1ifa=0 ANDb=1OR ifa= 1 ANDb=0OR ifa=1 ANDb=1 f is 1ifa=1 ANDb=1OR ifa= 1 ANDb=1OR ifa=1 ANDb=1 f is 1if ab = 1 OR if ab = 1 OR if ab = 1 f = ab + ab + ab

P11a.(a + b) = ab P11b.(ab) = a + b

Figure 2.9 NAND gates. Figure 2.10 Alternate symbol for NAND. Figure 2.11 Symbols for NOR gate.

Figure 2.13 NAND gate implementation.

= ab + ab Figure 2.17 An Exclusive OR gate.

For any two product terms where exactly one variable appears uncomplemented in one and complemented in the other, the consensus is defined as the product of the remaining literals. If no such variable exists or if more than one such variable exists, then the consensus is undefined. If we write one term as at 1 and the second as at 2 (where t 1 and t 2 represent product terms), then, if the consensus is defined. at 1 ¢ at 2 = t 1 t 2 P13a. at 1 + at 2 + t 1 t 2 = at 1 + at 2 P13b. (a + t 1 )(a + t 2 )(t 1 + t 2 )= (a + t 1 )(a + t 2 )

G = DE + ABC + CDE + ABCE G = C (AB + ABE) + DE + CDE G = C (B + AE)(B + A) + DE + CDE G = (C + D E)[C + (B + AE)(B + A)] + DE

Counting Binary DecimalHexadecimal To the chalk board...

More Examples Convert to binary and hexadecimal and octal. Convert to decimal an octal. Convert to binary. Convert to binary.

Exercises Add these binary numbers: What 2 kinds of logic gates are needed for computer addition?

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Chapter 2 Switching Algebra and Logic Circuits Remember to bring your book to class on Tuesday for Lab. Remember Homework 1 & 2.