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. Expressions
We can simplify boolean expressions
AB + AC A(B + C)

9. Simplification
Thus we can simplify circuits
AB + AC A(B + C)

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

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

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

13. 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

14. Implication 1 Can build circuit from SOP expression:
Any circuit can be build with NOT, AND & OR

15. SOP Circuit Can build circuit from SOP expression…
… but can often do better…

16. Simplification Can simplify things…
X = A B + A B + AB = A B + A( B + B) = A B + A(1) = A B + A = B + A = A + B

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

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

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

20. 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

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

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

23. Single Gate Logic Fun Fact
Any circuit can be build with NAND gates only…

24. Fun Fact
Single Gate Logic
Replace NOT with NAND

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

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

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