Topic 2 Subtopic 2: Boolean Equation Forms

Boolean algebra is a form of algebra that deals with single digit binary values and variables.
Values and variables can indicate some of the following binary pairs of
values:

• ON / OFF
• TRUE / FALSE
• HIGH / LOW
• CLOSED / OPEN
• 1 / 0


Basic Identities

X + 0 = X                                  X ⋅ 1 = X                                   Identity
X + 1 = 1                                   X ⋅ 0 = 0
X + X′ = 1                                 X ⋅ X′ = 0                                  Complement
(X′)′ = X                                                                                      Involution Law
X + Y = Y + X                          XY = YX                                     Commutativity
X + (Y + Z) = (X + Y) + Z        X(YZ) = (XY)Z                             Associativity
X(Y + Z) = XY + XZ                X + YZ = (X + Y)(X + Z)               Distributivity
X + X = X                                 X ⋅ X = X                                    Idempotent Law
X + XY = X                              X(X + Y) = X                               Absorption Law
X + X′Y = X + Y                      X(X′ + Y) = XY                             Simplification
(X + Y)′ = X′Y′                         (XY)′ = X′ + Y′                              DeMorgan’s Law
XY + X′Z + YZ                         (X + Y) (X′ + Z)(Y + Z)                 Consensus Theorem
= XY + X′Z                               = (X + Y)(X′ + Z)

Boolean expressions can be manipulated into many forms.
Some standardized forms are required for Boolean expressions to simplify
communication of the expressions.
• Sum-of-products (SOP)
• Example:
• Products-of-sums (POS)
Example:

F(A, B, C, D) = AB + BCD + AD
F(A, B, C, D) = (A + B)(B + C + D)(A + D)

Liew Jun Jie               B031210374