1. Hardware Implementations Gates and Circuits

2. Three Main Gates  AND  OR  NOT

3. Gate Diagrams  Example 1: [(today is Monday) AND (it is raining)] OR (it is snowing)

4. Gate Diagrams  Example: What does it represent?

5. Gate Diagrams  Example: {[(today is Monday) AND (it is raining)] OR (it is snowing)} AND {NOT [(it is raining) AND (it is snowing)]}

6. Truth Table to Gates  First, build the Boolean algebra expression that gives Z Z = AB + A’B’ Z = (A AND B) OR (NOT A AND NOT B) ABZ TTT TFF FTF FFT

7. Truth Table to Gates  Z = AB + A’B’  Next, build the circuit that goes with the Boolean algebra expression Z ABZ TTT TFF FTF FFT

8. Z = AB + A’B’

9. Binary Arithmetic  We can add binary numbers just like decimal numbers only using base two arithmetic.  For example: 5101 1110 101 + 7+ 111 121100

10. Binary Addition  Notice in addition: 0011 + 0+ 1+ 0 + 1 01110 FalseTrue False True Sum Carry ABSum (1) T (0) F (1) T(0) F(1) T (0) F(1) T (0) F

11. Sum and Carry ABCarry 111 100 010 000 ABSum 110 101 011 000

12. Sum Circuit ABSum 110 101 011 000 Sum = AB’ + A’B

13. Carry Circuit ABCarry 111 100 010 000 Carry = AB

14. Half Adder - Sum and Carry

15. Half Adder  The sum digit is 0 if the sum is even.  The sum digit is 1 if the sum is odd.  The carry is 1 if the sum is greater than 1.  Handles the case where we add two binary digits with no inward carry.

16. Full Adder  Takes a carry in and produces the result and carry out.  So, we have 3 inputs and two outputs.  Combine two half-adders together with an OR gate to get a full adder for each binary digit.  How many half adders would we need to add two 8-digit binary numbers? How many gates?

17. Full Adder

18. Subtraction ABSub 110 101 011 000 ABBorrow 110 100 011 000

19. Binary Subtraction  We do binary subtraction like decimal subtraction only the borrowing is done in 2’s instead of 10’s. 12201111010 - 7- 00000111 11501110011

20. Subtraction as Addition  If A = 01111010, B = 00000111, then using the twos-complement representation for –B, we have –B = 11111000 + 1 = 11111001 so 12201111010 - 7+ 11111001 11501110011

21. Binary Multiplication  Again, just like decimal except we add and multiply in binary. *01 000 101 5101 x 7x 111 35100011

22. NAND Gates and NOT  This gate represents (A NAND NOT B).

23. NAND Truth Table ABA NAND B TTF TFT FTT FFT

24. NAND  Fact: All other gates (AND, OR, NOT) can be constructed using only NAND gates  Verification:

25. Exercises  Fill in a truth table and give a Boolean expression for the following circuits.

26. Exercises - How would you create a one binary digit multiplier? A two-digit by one-digit multiplier? A two-digit by two-digit multiplier? *01 000 101