1. Circuits, Truth Tables & Boolean Algebra

2. Expressions
Can describe circuits in terms of Boolean expression

3. Expressions
Write Q as a function of A, B and C

4. Expressions
Write Q as a function of A, B and C

5. Expressions
Write Q as a function of A, B and C

6. Expressions
Write Q as a function of A, B and C

7. Expressions
Can represent a circuit's output with Boolean expression:
AB + AC

8. Sum Of Products Can convert any truth table to expression
Identify ways to get 1 output
X = 1 if red OR green OR yellow

9. Sum Of Products
Sum of Products : Individual terms are products Terms OR’d together
X = A B + A B + AB

10. Multiple Inputs Each product shows EVERY input
Either raw or inverted
Ex: Each product has A or A B or B C or C

11. SOP Circuit Can build circuit from SOP expression: X = A B + A B + AB
Inputs feed layer of AND gates
Some inputs inverted first
OR gate takes ANDs
X = A B + A B + AB

12. Multiple Outputs Multiple outputs
Each output separate function/circuit IN
OUT A B X Y 1

13. Multiple Outputs Multiple outputs X = AB
Each output separate function/circuit
X = AB IN
OUT A B X Y 1

14. Multiple Outputs Multiple outputs X = AB Y = AB + A B
Each output separate function/circuit
X = AB
Y = AB + A B IN
OUT A B X Y 1

15. Logisim Build simplified circuit from truth table…
Window  Combinational Analysis
Build truth table for circuit…
Project  Analyze Circuit

16. Multiple Outputs Multiple outputs X = AB Y = AB + A B
Each output separate function/circuit
X = AB
Y = AB + A B

17. SOP Circuit Fun Fact Can build circuit from SOP expression…
… but can often do better…

18. Expressions AB + AC A(B + C) Fun Fact
We can simplify Boolean expressions
AB + AC A(B + C)

19. Fun Fact
Simplification
Thus we can simplify circuits
AB + AC A(B + C)

20. Simplification Fun Fact May be able to simplify SOP
X = A B + A B + AB = A B + A( B + B) = A B + A(1) = A B + A = B + A = A + B

21. Single Gate Logic Fun Fact Can make NOT, AND & OR with only NAND

22. Single Gate Logic Fun Fact Can build circuit from SOP expression:
Any circuit can be build with NOT, AND & OR

23. Fun Fact
Single Gate Logic
Replace NOT with NAND

24. Single Gate Logic Fun Fact Put inversion bubbles out of AND & into OR
2 negations cancel out

25. Single Gate Logic Fun Fact Convert A + B + …. to AB...
De Morgan's Theorem

26. Single Gate Logic Fun Fact Circuit built entirely with NAND
Can do the same trick with NOR