• 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).
    3. Add 1.

    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