MATH 1310 (2014)

Basic Postulates and Theorems of Boolean Algebra
The following postulates, properties, theorems, laws, of Boolean algebra will form the standard list for use in the course, along with the standard (for this course) identifying label in each case.
Here x, y and z represent logical variables.

Postulates:
P1a

x = 1 if x  0

P1b

x = 0 if x  1

P2a
P3a
P4a
P5a

00 = 0
11 = 1
10 = 0

P2b
P3b
P4b
P5b

0+0=0
1+1=1
1+0=1

Commutativity:
P6a
xy = yx

P6b

x+y=y+x

Associativity:
P7a
x(yz) = (xy)z

P7b

x + (y + z) = (x + y) + z

Distributivity:
P8a
x(y + z) = xy + xz

P8b

x + yz = (x + y)(x + z)

1  1  0

0  0  1

Algebraic Properties

Theorems
T9a
T10a

x0 = 0 x1 = x

T9b x + 0 = x
T10b x + 1 = 1

Idempotency:
T11a
xx = x

T11b x + x = x

Inverse Elements:
T12a
x•x = x•x = 0

T12b

x + x = x + x = 1

x = x = x

T13b

x = x = x

T14a

x + xy = x

T14b x(x + y) = x

T14c

x x + y = x•y

T14d

x + x•y = x + y

T15b

x + y + z = x•y•z

Double Negation Theorems:
T13a
Absorption





DeMorgan’s Theorems
T15a

x•y•z = x + y + z

Math 1310 (2014)

Basic Logic Operations, Truth Tables and Digital Logic Gates
The following table gives the eight distinct basic logic operations we will consider in this course, with their truth tables, usual algebraic symbols, and the graphic symbol for the associated digital logic gate. In this table, x, y, and Z are logical variables/functions.
Name

AND

OR

NOT

Algebraic Symbols

Z = x AND y
Z = xy = xy =