• BME 210
• # Handout 10: Boolean Algebra

## Binary Number Representation

Two’s complement, or how to figure out the binary representation of a decimal number:

1. Write down the unsigned, positive binary representation. The MSB is where the “sign” will go and it will be 0.
2. Complement each of the bits (change all 0’s to 1’s and vice versa).

## Truth Tables for Basic Logical Operations

### AND

$$X$$ $$Y$$ $$Z=XY$$
0 0 0
0 1 0
1 0 0
1 1 1

### OR

$$X$$ $$Y$$ $$Z=X+Y$$
0 0 0
0 1 1
1 0 1
1 1 1

### NOT

$$X$$ $$Z=\overline{X}$$
0 1
1 0

### NAND

$$X$$ $$Y$$ $$Z=\overline{XY}$$
0 0 1
0 1 1
1 0 1
1 1 0

### NOR

$$X$$ $$Y$$ $$Z=\overline{X+Y}$$
0 0 1
0 1 0
1 0 0
1 1 0

### XOR

$$X$$ $$Y$$ $$Z=X\oplus Y$$
0 0 0
0 1 1
1 0 1
1 1 0

## Basic Boolean Identities

1. $$X+0=X$$
2. $$X1=X$$
3. $$X+1=1$$
4. $$X0=0$$
5. $$X+X=X$$
6. $$XX=X$$
7. $$X+\overline{X}=1$$
8. $$X\overline{X}=0$$
9. $$\overline{\overline{X}}=X$$

10. $$X+Y=Y+X$$

11. $$XY=YX$$

12. $$X+(Y+Z)=(X+Y)+Z$$

13. $$X(YZ)=(XY)Z$$

14. $$X(Y+Z)=XY+XZ$$

15. $$X+YZ=(X+Y)(X+Z)$$

16. $$\overline{X+Y}=\overline{X}\overline{Y}$$

17. $$\overline{XY}=\overline{X}+\overline{Y}$$

Last updated:
January 6, 2018