Binary Logic, Gates, and Boolean Algebra
Tom Kelliher, CS 240
Jan. 27, 2012
Read 2.3-4.
Introduction
- Binary logic and gates.
- Boolean Algebra.
Standard forms, maps, and minimization.
- Fundamental operators and their symbols:
- AND
- OR
- NOT
- NAND is complete.
- Gate fan-in and fan-out. Electrical significance.
- Timing diagram.
- Frequency and period.
- Timing diagrams. Show AND, OR, NOT waveforms for input: A: 0011,
B: 0101.
- What do the waveforms really look like: propagation delay, noise,
under- and over-shoot.
- Boolean functions can be represented by equations, truth tables, or
logic circuits.
How do you convert from one form to another?
How many rows in the truth table of an
-input Boolean function?
- Why would we want to simplify a Boolean equation?
- Basic Identities:
1. ![$X + 0 = X$](jan27img2.png) |
2. ![$X \cdot 1 = X$](jan27img3.png) |
3. ![$X + 1 = 1$](jan27img4.png) |
4. ![$X \cdot 0 = 0$](jan27img5.png) |
5. ![$X + X = X$](jan27img6.png) |
6. ![$X \cdot X = X$](jan27img7.png) |
7.
![$X + \overline{X} = 1$](jan27img8.png) |
8.
![$X \cdot \overline{X} = 0$](jan27img9.png) |
9.
![$\overline{\overline{X}} = X$](jan27img10.png) |
|
10. ![$X + Y = Y + X$](jan27img11.png) |
11. ![$XY = YX$](jan27img12.png) |
12.
![$X + (Y + Z) = (X + Y) + Z$](jan27img13.png) |
13. ![$X(YZ) = (XY)Z$](jan27img14.png) |
14.
![$X(Y + Z) = XY + XZ$](jan27img15.png) |
15.
![$X + YZ = (X + Y)(X + Z)$](jan27img16.png) |
16.
![$\overline{X + Y} = \overline{X} \cdot \overline{Y}$](jan27img17.png) |
17.
![$\overline{X \cdot Y} = \overline{X} + \overline{Y}$](jan27img18.png) |
Example simplifications. Use both a truth table and Boolean manipulation
to show:
Thomas P. Kelliher
2012-01-25
Tom Kelliher