INRODUCTION TO BOOLEAN ALGEBRA !

•   In the year 1854, George Boole, an English mathematician, proposed this algebra.
• This is a variant of Aristotle’s propositional logic that uses the symbols 0 and 1, or True and False. Boolean algebra is concerned with binary variables and logic operations.
•  It is fundamental to digital logic design and computer science, providing a mathematical framework for describing logical operations and expressions.

• Various operations are used in Boolean algebra but the basic operations that form the base of Boolean Algebra are.

  • Negation or NOT Operation
  • Conjunction or AND Operation
  • Disjunction or OR Operation

Negation or NOT Operation

• Using the NOT operation reverse the value of the Boolean variable from 0 to 1 or vice-versa. This can be understood as:

  • If A = 1, then using NOT operation we have (A)’ = 0

  • If A = 0, then using the NOT operation we have (A)’ = 1
  • We also represent the negation operation as ~A, i.e if A = 1, ~A = 0

Conjunction or AND Operation

• Using the AND operation satisfies the condition if both the value of the individual variables are true and if any of the value is false then this operation gives the negative result. This can be understood as,

  • If A = True, B = True, then A . B = True

  • If A = True, B = False, Or A = false, B = True, then A . B = False

  • If A = False, B = False, then A . B = False

Disjunction (OR) Operation

Using the OR operation satisfies the condition if any value of the individual variables is true, it only gives a negative result if both the values are false. This can be understood as,

• If A = True, B = True, then A + B = True

• If A = True, B = False, Or A = false, B = True, then A + B = True

• If A = False, B = False, then A + B = Falses

LAWS FOR BOOLEAN ALGEBRA

• The basic laws of the Boolean Algebra are added in the table added below:

  Law	OR form	AND form
  Identity Law	P + 0 = P	P.1 = P
  Idempotent Law	P + P = P	P.P = P
  Commutative Law	P + Q = Q + P	P.Q = Q.P
  Associative Law	P + (Q + R) = (P + Q) + R	P.(Q.R) = (P.Q).R
  Distributive Law	P + QR = (P + Q).(P + R)	P.(Q + R) = P.Q + P.R
  Inversion Law	(A’)’ = A	(A’)’ = A
  De Morgan’s Law	(P + Q)’ = (P)’.(Q)’	(P.Q)’ = (P)’ + (Q)’

TRUTH TABLE IN BOOLEAN ALGEBRA

• A truth table represents all the combinations of input values and outputs in a tabular manner. All the possibilities of the input and output are shown in it and hence the name truth table.
•  In logic problems, truth tables are commonly used to represent various cases. T or 1 denotes ‘True’ & F or 0 denotes ‘False’ in the truth table.

BOOLEAN ALGEBRA THEOREM

• There are two basic theorems of great importance in Boolean Algebra, which are De Morgan’s First Laws, and De Morgan’s Second Laws. These are also called De Morgan’s Theorems.

DE MORGAN'S FIRST LAW

• It states that the complement of the product (AND) of two Boolean variables (or expressions) is equal to the sum (OR) of the complement of each Boolean variable (or expression).

   (P.Q)’ = (P)’ + (Q)’

• The truth table for the same is given below:

  P	Q	(P)’	(Q)’	(P.Q)’	(P)’ + (Q)’
  T	T	F	F	F	F
  T	F	F	T	T	T
  F	T	T	F	T	T
  F	F	T	T	T	T

DE MORGAN'S SECOND LAW

• Statement: The Complement of the sum (OR) of two Boolean variables (or expressions) is equal to the product(AND) of the complement of each Boolean variable (or expression).

  (P + Q)’ = (P)’.(Q)’

• The truth table for the same is given below:

  P	Q	(P)’	(Q)’	(P + Q)’	(P)’.(Q)’
  T	T	F	F	F	F
  T	F	F	T	F	F
  F	T	T	F	F	F
  F	F	T	T	T	T

POSTULATES IN BOOLEAN ALGEBRA

• A postulate is a statement that is presumed to be true when it hasn’t been proven. Postulates in Boolean algebra are not laws and theorems but statements that hold true. 

• The four potential logical OR and logical AND operations, as well as the rules followed by the NOT operator, are included in these postulates. The postulates in Boolean algebra is given below:

  • 0 + 0 = 0
  • 0 + 1 = 1
  • 1 + 0 = 1
  • 1 + 1 = 1
  • 0 . 0 = 0
  • 0 . 1 = 0
  • 1 . 0 = 0
  • 1 . 1 = 1
  • 1′ = 0
  • 0′ = 1

LOGIC GATES

• Logic Gates are the fundamental building blocks in digital electronics.
• A logic gates are an electronic circuit that are designed by using electrical components like diodes, transistors, resistors and more.
• It is used to perform logical operations based on the inputs provided to it and gives logical output that can be either high(1) or low(0).

TYPES OF LOGIC GATES:

AND GATE

• An AND gate is used to perform logical Multiplication of binary input. The Output state of the AND gate will be high(1) if both the input are high(1) ,else the output state will be low(0) if any of the input is low(0).

• The Boolean Expression or logic for the AND gate is the logical multiplication of inputs denoted by a full stop or single dot as

                                A.B=X

  The value of X will be True when both the inputs will be True.

OR GATE

• OR gate is most widely used digital logic circuit. The output state of OR gate will be high i.e.,(1) if any of the input state is high or 1, else output state will be low i.e., 0.

• The Boolean Expression for the OR gate is the logical addition of inputs denoted by plus sign(+) as

  X= A+B

  The value of X will be high(true) when one of the inputs is set to high (true).

NOT GATE

• In digital electronics, the NOT gate is one of the basic logic gate having only a single input and a single output. It is also known as inverter or inverting buffer. When the input signal is “low” the output signal is “high” and vice-versa.

• The Boolean expression of NOT Gate is as follows

          Y = Ā or Y = A’

  The value of Y will be high when A will be low.

NOR GATE

• The NOR gate is the type of universal logic gate. It takes two or more inputs and gives only one output. The output state of the NOR gate will be high(1) when all the inputs are low(0). NOR gate returns the complement result of the OR gate. It is basically a combination of two basic logic gates i.e., OR gate and NOT gate.

• The Boolean expression of NOR gate is as follows :

  If A and B are considered as two inputs, and O as output, then the expression for a two input NOR gate will be

                O = (A + B)’

  The value of O will be true when all of its inputs are set to 0.

NAND GATE

• The NAND gate or “Not AND” is the combination of two basic logic gates AND gate and the NOT gate connected in series. It takes two or more inputs and gives only one output. The output of the NAND gate will give result high(1) when either of its input is high(1) or both of its input are low(0).

• The Boolean Expression of NAND Gate is as follows:

  Say we have two inputs, A and B and the output is called X, then the expression is

        X = (A . B)’

XOR GATE

• In digital electronics, there is a specially designed logic gate named, XOR gate, which is used in digital circuits to perform modulo sum.
• The XOR gate can take only two inputs at a time and give an output. The output of the XOR gate is high(1) only when its two inputs are dissimilar i.e., if one of them is low(0) then other one will be high(1).

• The Boolean expression of XOR Gate is as follows:

  X = A’B + AB’

XNOR GATE

• The XNOR is the combination of XOR gate and NOT gate. The output of the XNOR gate is high(1) when both the inputs are high(1) or low(0).
• The XNOR gate is the complement of the XOR gate.

• The following is the Boolean expression of the XNOR gate:

  Y = A ⊙ B

• This expression can also be written as follows:

  Y = AB + A’B’