Operations with 0 and 1

• X + 0 = X
• X + 1 = 1
• X . 0 = 0
• X . 1 = X

Idempotent Laws

• X + X = X
• X . X = X

Involution Law

• ( X' )' = X

Complementarity Law

• X + X' = 1
• X . X' = 0

Commutative Laws

• X + Y = Y + X
• X . Y = Y . X

Associative Laws

• ( X + Y ) + Z = X + ( Y + Z ) = X + Y + Z
• ( X . Y ) . Z = X . ( Y . Z ) = X . Y . Z

Distributive Laws

• X . ( Y + Z ) = ( X . Y ) + ( X . Z )
• X + ( Y . Z ) = ( X + Y ) . ( X + Z )

Simplification Theorems

• ( X . Y ) + ( X . Y' ) = X
• ( X + Y ) . ( X + Y' ) = X
• X + ( X . Y ) = X
• X . ( X + Y ) = X
• ( X + Y' ) . Y = X . Y
• ( X . Y' ) + Y = X + Y

DeMorgan's Laws

• ( X + Y )' = X' . Y'
• ( X . Y )' = X' + Y'

Theorem for Multiplying and Factoring

• ( X + Y ) . ( X' + Z ) = ( X . Z ) + ( X' . Y )
• ( X . Y ) + ( X' . Z ) = ( X + Z ) . ( X' + Y )

Consensus Theorem

• ( X . Y ) + ( Y . Z ) + ( X' . Z ) = ( X . Y ) + ( X' . Z )
• ( X + Y ) . ( Y + Z ) . ( X' + Z ) = ( X + Y ) . ( X' + Z )