Boolean Algebra

Truth Tables for the Laws of Boolean

Boolean Expression	Description	Equivalent Switching Circuit	Boolean Algebra Law or Rule
A + 1 = 1	A in parallel with closed = "CLOSED"		Annulment
A + 0 = A	A in parallel with open = "A"		Identity
A . 1 = A	A in series with closed = "A"		Identity
A . 0 = 0	A in series with open = "OPEN"		Annulment
A + A = A	A in parallel with A = "A"		Idempotent
A . A = A	A in series with A = "A"		Idempotent
NOT A = A	NOT NOT A (double negative) = "A"		Double Negation
A + A = 1	A in parallel with NOT A = "CLOSED"		Complement
A . A = 0	A in series with NOT A = "OPEN"		Complement
A+B = B+A	A in parallel with B = B in parallel with A		Commutative
A.B = B.A	A in series with B = B in series with A		Commutative
A+B = A.B	invert and replace OR with AND		de Morgan’s Theorem
A.B = A+B	invert and replace AND with OR		de Morgan’s Theorem

Description of the Laws of Boolean Algebra

• Annulment Law – A term AND´ed with a “0” equals 0 or OR´ed with a “1” will equal 1.
•  
  • A . 0 = 0    A variable AND’ed with 0 is always equal to 0.
  • A + 1 = 1    A variable OR’ed with 1 is always equal to 1.
•  
• Identity Law – A term OR´ed with a “0” or AND´ed with a “1” will always equal that term.
•  
  • A + 0 = A   A variable OR’ed with 0 is always equal to the variable.
  • A . 1 = A    A variable AND’ed with 1 is always equal to the variable.
•  
• Idempotent Law – An input that is AND´ed or OR´ed with itself is equal to that input.
•  
  • A + A = A    A variable OR’ed with itself is always equal to the variable.
  • A . A = A    A variable AND’ed with itself is always equal to the variable.
•  
• Complement Law – A term AND´ed with its complement equals “0” and a term OR´ed with its complement equals “1”.
•  
  • A . A = 0    A variable AND’ed with its complement is always equal to 0.
  • A + A = 1    A variable OR’ed with its complement is always equal to 1.
•  
• Commutative Law – The order of application of two separate terms is not important.
•  
  • A . B = B . A    The order in which two variables are AND’ed makes no difference.
  • A + B = B + A    The order in which two variables are OR’ed makes no difference.
•  
• Double Negation Law – A term that is inverted twice is equal to the original term.
•  
  • A = A     A double complement of a variable is always equal to the variable.
•  
• de Morgan´s Theorem – There are two “de Morgan´s” rules or theorems,
•  
• (1) Two separate terms NOR´ed together is the same as the two terms inverted (Complement) and AND´ed for example, A+B = A. B.
•  
• (2) Two separate terms NAND´ed together is the same as the two terms inverted (Complement) and OR´ed for example, A.B = A +B.

 

Other algebraic Laws of Boolean not detailed above include:

• Distributive Law – This law permits the multiplying or factoring out of an expression.
•  
  • A(B + C) = A.B + A.C    (OR Distributive Law)
  • A + (B.C) = (A + B).(A + C)    (AND Distributive Law)
•  
• Absorptive Law – This law enables a reduction in a complicated expression to a simpler one by absorbing like terms.
•  
  • A + (A.B) = A    (OR Absorption Law)
  • A(A + B) = A    (AND Absorption Law)
•  
• Associative Law – This law allows the removal of brackets from an expression and regrouping of the variables.
•  
• A + (B + C) = (A + B) + C = A + B + C    (OR Associate Law)
• A(B.C) = (A.B)C = A . B . C    (AND Associate Law)