1. CPSC 171 Introduction to Computer Science Boolean Logic, Gates, & Circuits

2. Announcements Read Chapter 4 Exam, Oct 2 nd in class

3. Boolean Logic A Boolean variable, A, is either true or false A Boolean expression, (A AND B), evaluates to either true or false Boolean operators include: AND (&  ) OR (  + ) NOT (a bar ' ¬ ~)

4. Boolean Operators a AND b true only when A and B are both true a OR b true when A is true, B is true, or both are true NOT a true when A is false

5. Truth Tables Truth tables can be used to capture when an expression is true, given its inputs aba AND b 001 011 101 111 You make truth tables for AND and NOT

6. Example Boolean Expressions (a AND b) OR (NOT a AND c) a·b + ~a·c ab+āc Truth tables can be made for complex expressions as well

7. abValue 001 010 100 111 Example: (a AND b) OR ((NOT b) and (NOT a)) Boolean Logic (continued)

8. Gates Hardware devices built from transistors to mimic Boolean logic An electronic device that operates on a collection of binary inputs to produce a single binary output AND gate (page 161 in text) Two input lines, one output line Outputs a 1 when both inputs are 1

9. Gates (continued) OR gate (page 163 in text) Two input lines, one output line Outputs a 1 when either input is 1 NOT gate (page 161 in text One input line, one output line Outputs a 1 when input is 0 and vice versa

10. Figure 4.15 The Three Basic Gates and Their Symbols

11. Circuits A collection of logic gates that transforms a set of binary inputs into a set of binary outputs Wire gates together keeping constraints for the number of inputs to any gate

12. Example Circuit If a, b, c, and d are all true the output can be determined by tracing through the circuit a b c d output 1 1 1 1 1 1 0 0

13. Designing Circuits A circuit construction algorithm 1.Truth Table Construction Determine outputs for every possible input 2.Sub-expression Construction (using AND and NOT gates) For each output find the rows that are 1 and build a sub- expression that is true for the exact input 3.Sub-expression combination (using OR gates) Take each subexpression and combine them, 2 at a time, using OR gates 4.Circuit Diagram Production Construct final circuit by converting Boolean operators into gates

14. Example Circuit Design Design a 3-input circuit that is true if exactly two inputs are true, and false otherwise You Try it: Design a 2-input circuit that is true if the inputs are the same, and false otherwise

15. Examples of Circuit Design and Construction Compare-for-equality circuit Addition circuit Both circuits can be built using the circuit design algorithm

16. CE compares two unsigned binary integers for equality Built by combining together 1-bit comparison circuits (1-CE) Integers are equal if corresponding bits are equal (AND together 1-CD circuits for each pair of bits) A Compare-for-Equality Circuit

17. 1-CE circuit truth table abOutput 001 010 100 111 A Compare-for-Equality Circuit (continued)

18. 1-CE Boolean expression First case: (NOT a) AND (NOT b) Second case: a AND b Combined: ((NOT a) AND (NOT b)) OR (a AND b) A Compare-for-Equality Circuit (continued)

19. Figure 4.22 One-Bit Compare-for-Equality Circuit

20. N-Bit Compare for Equality Circuit AND together the 1-CE circuits, two at a time

21. An Addition Circuit Adds two unsigned binary integers, setting output bits and an overflow Built from 1-bit adders (1-ADD) Starting with rightmost bits, each pair produces A value for that order A carry bit for next place to the left

22. 1-ADD truth table Input  One bit from each input integer  One carry bit (always zero for rightmost bit) Output  One bit for output place value  One carry bit An Addition Circuit (continued)

23. Figure 4.24 The 1-ADD Circuit and Truth Table

24. Building the full adder Put rightmost bits into 1-ADD, with zero for the input carry Send 1-ADD’s output value to output, and put its carry value as input to 1-ADD for next bits to left Repeat process for all bits See pg 174, 175, 176 An Addition Circuit (continued)