1. 5 - Digital Logic with Boolean Algebra

2. Boolean Algebra AND Multiplication You have seen this as p^q
Usually Represented as AB or A*B A B AB 1

3. Boolean Algebra OR Addition You have seen this as pvq
Commonly Represented as A+B A B
A+B 1

4. Boolean Algebra NOT Inverse You have seen this as ~p or ~q
Usually Represented as A’ or A͞ A A’ 1

5. Properties Basic Properties Double Negative Law Complement Law A+1=1
A+A=A
A*A=A
Double Negative Law
(A’)’=A
Complement Law
A’+A=1
A*A’=0

6. Properties Law of Common Identities DeMorgan’s Theorem A*(A’+B)=AB

7. Boolean Algebra to Logic Gates
Logic circuits are built from components called logic gates.
The logic gates correspond to Boolean operations +, *, ’.
Binary operations have two inputs, unary has one OR +
AND *
NOT ’

8. The AND Gate:
Truth Table: The table shows that the AND gate responds with a high at the output if the signal applied to the input A and B are both high.
Animation: In order to see how it works, the gate has been connected to 2 switches and LED. Continue to see the system in action… 5v
Logic 1
Logic 0
Logic 1
Input A
Output X
Input B
AND
Logic 0
Logic 1
Logic 0 A B X 1

9. AND A
Logic Gate:
A*B
Truth Table: B A B
A*B 1 A B
Series Circuit:
A*B

10. The OR Gate:
Truth Table: The table shows that the OR gate responds with a high at the output if the signal applied to the input A or B is high.
Animation: In order to see how it works, the gate has been connected to 2 switches and LED. Continue to see the system in action… 5v
Logic 1
Logic 1
Logic 0
Input A
Output X
Input B OR
Logic 0
Logic 1
Logic 0 A B X 1

11. OR A Logic Gate: A+B Truth Table: B A B A+B 1 A Parallel Circuit: B1 A
Parallel Circuit: B
A+B

12. The NOT Gate (inverter):X 1
Truth Table: Is a chart that lists the input condition on the left and the gate’s output response on the right. The table shows that the NOT gate responds at the output with the inverse of the signal applied to the input.
Animation: In order to see how it works, the gate has been connected to a switch and LED. Continue to see the system in action… 5v
Input A
Output X
Logic 1
OFF
Logic 0 ON
Logic 0 ON
Logic 0 ON
Logic 1
OFF
Logic 1
OFF

13. n-input Gates
Because + and * are binary operations, they can be cascaded together to OR or AND multiple inputs. A A B
A+B+C
ABC B C A A B
A+B+C
ABC B C C

14. n-bit Inputs
For convenience, it is sometimes useful to think of the logic gates processing n-bits at a time. This really refers to n instances of the logic gate, not a single logic date with n-inputs.
110001
001110

15. Logic Circuits ≡ Boolean Expressions
All logic circuits are equivalent to Boolean expressions and any boolean expression can be rendered as a logic circuit.
AND-OR logic circuits are equivalent to sum-of-products form.
Consider the following circuits: A
y=aB+Bc
abc B C A B C Y
aBc y Ab
y=abc+aBc+Ab

16. NAND and NOR Gates
NAND and NOR gates can greatly simplify circuit diagrams. As we will see, can you use these gates wherever you could use AND, OR, and NOT. A B
AB 1
NAND A B
AB 1
NOR

17. XOR and XNOR Gates
XOR is used to choose between two mutually exclusive inputs. Unlike OR, XOR is true only when one input or the other is true, not both. A B
AB 1
XOR A B
A B 1
XNOR

18. NAND and NOR as Universal Logic Gates
Any logic circuit can be built using only NAND gates, or only NOR gates. They are the only logic gate needed.
Here are the NAND equivalents:

19. NAND and NOR as Universal Logic Gates (cont)
Here are the NOR equivalents:
NAND and NOR can be used to reduce the number of required gates in a circuit.

20. Practice Problems
Predict the simplified output & develop a Truth Table: 2.