Number System Conversion and Boolean Logic : Class 11 NEB Notes and Numerical Solutions
Get Computer Science Number system conversion and boolean logic comprehensive notes, detailed Q&As, and past question papers solutions and video tutorials.
Number system refers to the base, arrangements and positional value of digits in number system.
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INTRODUCTIONTO NUMBER SYSTEM
In every step of life, we definitely find the use of numbers. If you need to buy something, you will have to pay a certain amount of money for which you will have to count the money. Likewise, the shopkeeper will also count the goods to give you and same for your changes. Therefore, number system can simply be defined as a way to represent numbers.
TYPESOF NUMBER SYSTEM
1) Decimal Number System:
The decimal number system is the most commonly used number system in our daily life. This generally used number system is also known as the base 10 number system because it uses just the 10 symbols i.e. 0 to 9. It is also known as the denary number system because any numeric value there is, these system’s digits can easily represent them.The decimal system is specially used in the computer interface. The weight and position of the digit dictate the value represented by it.
In this system, each number consists of the digits that are located at different positions. The positions of the 1^{st} and the 2^{nd} digits towards the right side of the decimal point are ^{1} and ^{2.} Similarly, the positions of the 1^{st} and 2^{nd} digits towards the left side of the decimal point are 0 and 1 respectively.
The value of the number is determined by adding the results out of the multiplication of the digits with the weight of their position. This method is called the expansion method. Under this method, the rightmost digit of the number is called the Least Significant Digit (LSD), as it has the lowest weight. Likewise, the leftmost digit of the number is called the Most Significant Digit (MSD), as it has the highest weight.
Examples:
(a) The weights and positions of each digit of the number 796 are as follows:
Positions 
2 
1 
0 
Weights 
10^{2} 
10^{1} 
10^{0} 
Face value 
7 
9 
6 
The above table indicates that:
The value of digit 7 = 7x10^{2}=700
The value of digit 9 = 9x10^{1}=90
The value of digit 6 = 6x10^{0}=6
The actual number can be formed by adding the values obtained by the digits as follows:
700+90+6=796
Here, the digit 7 in the number 796 is the most significant digit and 6 is the least significant digit.
(b) The weights and positions of each digit of the number 125.64 are as follows:
Positions 
2 
1 
0 
1 
2 
Weights 
10^{2} 
10^{1} 
10^{0} 
10^{1} 
10^{2} 
Face value 
1 
2 
5 
6 
4 
The above table indicates that:
The value of digit 1 = 1x10^{2} = 100
The value of digit 2 = 2x10^{1} = 20
The value of digit 5 = 5x10^{0}= 5
The value of digit 6 = 6x10^{1}= 0.6
The value of digit 4 = 4x10^{2}= 0.4
The actual number can be formed by adding the values obtained by the digits as follows:
100+20+5+0.6+0.4=125.64
2) Binary Number System:
Binary Number System refers to the number system that uses only two symbols i.e. 0 and 1, that's 0why it is called a base 2 number system. It is also known as Binary Digit (BIT). This number system is especially used in the internal processing of computer system. When we count up from 0 in binary, symbols are much more frequently run out as this system only uses 0 and 1 and 2 do not exist. Therefore, we use 10 in this system because 10 is equal to 2 in decimal. The combination of binary numbers can be used to represent different quantities like 1001. In Binary, each digit’s positional value is twice the face value or place value of the digit of its right side. Each position’s weight is a power of 2.
According to the position and weight, the place value of the digits is as follows:
Positions 
3 
2 
1 
0 
Weights 
2^{3} 
2^{2} 
2^{1} 
2^{0} 
Example: Convert 101.11
Positions 
2 
1 
0 
1 
2 
Face value 
1 
0 
1 
1 
1 
Weight 
2^{3} 
2¹ 
2^{0} 
2^{1} 
2^{2} 
The above table indicates that:
101.101 = 1x2²+0x2¹+1x2^{0}+1x2^{1}+1x2^{2}
= 1x4+0+1x1+1/2+1/4
= 4+0+1+0.5+0.25
= 5.75
3) Octal Number System:
Octal Number System is the base 8 system. Like the decimal number system, this system is also used in the internal processing of computer system. It is the system that consists of eight digits i.e. {0, 1, 2, 3, 4, 5, 6, 7} which is used for the representation of long binary numbers shorthandedly. In this system, each digit position represents a power of 8. The number 708 will not be valid in this system as 8 is not a valid digit.
According to the position and weight, the place value of the digits is as follows:
Example: Convert 128 to a decimal number.
Positions 
4 
3 
2 
1 
0 
Weights 
8^{4} 
8³ 
8² 
8¹ 
8^{0} 
128 = 1x8¹+2x8^{0}
= 1x8+2x1
= 8+2
= 10
4) Hexadecimal NumberSystem:
Hexadecimal Number System is the number system that represents long binary numbers in shortcut method. It is a base 16 system as it consists of 16 digits i.e. {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F } where the alphabets represent the decimal numbers 10 to 15. This system is also used in the computer system, mainly in the memory management. As the name suggests, each digit's position represents a power of 16 in this system.
According to the position and weight, the place value of the digits is as follows:
Example: Convert 2A16 to a decimal number.
Positions 
4 
3 
2 
1 
0 
Weights 
16^{4} 
16³ 
16² 
16¹ 
16^{0} 
2A16 = 2x16¹+Ax16^{0}
= 2x16+10x1
= 32+10
= 42
Number system conversion: binary, octal and hexadecimal
CONVERSION OF BINARY
It consists of :
 conversion from binary to decimalnumber system
 conversion from binary to octal number system
 conversion from binary to hexadecimal number system
Conversion of binary to decimal (base 2 to base 10):
The rules for conversion from binary to decimal are given below:
 Multiply each bit by corresponding power of 2 (base).
 Sum each product term to get a decimal equivalent number.
Note: A power of 2 is 0 for a left bit of binary point (or for a right most bit for the number that does not contain fractional part) and increase the power by one for each bit towards left and decrease power by one towards the right of binary point.
Example 1: convert (110011)_{2} to decimal.
Solution:
(110011)_{2 }= 1×2^{5} + 1×2^{4} + 0×2^{3} + 0×2^{2} + 1×2^{1} + 1×2^{0}
= 32 + 16 + 0 + 0 + 2 +1
= (51)_{10}
Example 2: convert (1011.101)_{2} into decimal.
(1011.101)_{2 }= 1×2^{3} + 0×2^{2} + 1×2^{1} + 1×2^{0} + 1×2^{1} + 0×2^{2} + 1×2^{3}
= 8 + 0 + 2 + 1 +0.5+ 0 +0.125
= 11 + 0.5 + 0.125
= (11.625)_{10}
Conversion of binary to octal (base 2 to base 8):
The rules for conversion from binary to decimal are given below:
 Divide the given binary number into a group of three bits from binary point to left (or from right to left if no fractional part) and from binary point to the right. Append 0’s at leading or trailing or trailing position if necessary to make each group of 3 bits.
 Substitute each group of three bits by octal equivalent from a table.
 Collect octal digits to get an octal equivalent number.
Example: convert (10110101)_{2 }into octal.
Solution:
Dividing the given binary number (10110101)_{2} into groups of 3 bits from right to left, and appending 0’s at leading position, we have,
Conversion of binary to hexadecimal (base 2 to base 16):
The rules for conversion from binary to hexadecimal are as given below:
 Divide the given binary number into a group of four bits from binary point to left (or from right to left if no fractional part) and from binary point to the right. Append 0’s at leading or trailing position if necessary to make each group of 4 bits.
 Substitute each group of four bits by hexadecimal equivalent symbol (letter or digit) from the table.
 Collect Hexadecimal symbols to get Hexadecimal equivalent number
Example: convert (10101011101)_{2} into hexadecimal
Solution:
Dividing the given binary number (10101011101)_{2} into groups of 4 bits from binary point to left and binary point right and appending 0’s at leading or at trailing position to make each group of four bits, we have,
Conversion of Decimal
It consists of conversion from decimal to binary, octal and hexadecimal number system which are described below:
Conversion of decimal to binary (base 10 to base 2):
The rules for conversion from decimal to binary are as given below:
 Divide the given number by 2 and note the remainder.
 Repeatedly divide the quotient by two and note the remainder until quotient reduced to 0.
 Collect the remainders, last obtained first and first obtained last to binary equivalent.
Example: convert (51)_{10 }into binary
Solution:
51÷2=25  remainder  = 1 
25÷ 2=12  remainder  = 1 
12÷ 2=6  remainder  = 0 
6÷ 2=3  remainder  = 0 
3÷ 2=1  remainder  = 1 
1÷ 2=0  remainder  = 1 
Therefore, (51)_{10 }= (110011)_{2}
Conversion of decimal to octal (base 10 to base 8):
The rules for conversion from decimal to binary are as given below:
 Divide the given number by 8 and note the remainder.
 Repeatedly divide the quotient by 8 and note the remainder until quotient reduced to 0.
 Collect the remainders, last obtained first and first obtained last to get the octal equivalent.
Example: convert (177)_{10} into octal
Solution:
177÷ 8 =22  remainder  = 1 
22÷ 8 = 2  remainder  = 6 
2÷ 8 = 0  remainder  = 2 
Therefore, (177)_{10} = (261)_{8}
Conversion of decimal to hexadecimal (base 10 to base 16):
Example: convert (77)_{10 }into hexadecimal
Solution:
77÷ 16=4  remainder  = 13 which means D 
4÷ 16 =0  remainder  = 4 
Therefore, (77)_{10 }= (4D)_{16}
Conversion of octal
It consists of conversion from octal to decimal, binary and hexadecimal number system which are described below:
Conversion of octal to decimal (base 8 to base 10):
Multiply the each octal digit by corresponding power of 8 and sum each product term to get decimal equivalent.
Example: convert (632)_{8} to decimal.
Solution:
(632)_{8} = 6×8^{2} + 3×8^{1} + 2×8^{0}
= 384 + 24 + 2
= 410
Therefore, (632)_{8} = (410)_{10}
Conversion from octal to binary (base 8 to base 2):
Substitute each octal digit by equivalent 3 bit binary from table and collect bits for each digit to get binary equivalent numbers.
Example: convert (741)_{8} into binary.
Conversion from octal to hexadecimal (base 8 to base 16):
Actually, there is no direct method for converting from octal to hexadecimal so first, convert octal into binary or decimal and then convert binary or decimal to hexadecimal.
Conversion of hexadecimal
It consists of conversion from hexadecimal to decimal, binary and octal number system which are described below:
Conversion of hexadecimal to decimal (base 16 to base 10):
The rules for conversion from hexadecimal to decimal are as given below:
 Multiply each digit by corresponding power of 16(base) as in decimal to binary.
 Sum each product term to get decimal equivalent.
Example: convert (F4C)_{16} into decimal.
Solution:
(F4C)_{16 }= F×16^{2} + 4×16^{1} + C×16^{0}
= 15×256 + 4×16 + 12×1
= 3840 + 64 + 12
= 3916
Therefore, (F4C)_{16 }= (3916)_{10}
Conversion of hexadecimal to binary (base 16 to base 2):
Substitute each hexadecimal symbol by equivalent 4 bit binary from table and collect bits for each digit to get binary equivalent numbers.
Conversion of hexadecimal to octal (base 16 to base 8):
There is no direct method for converting from hexadecimal to octal, so first convert hexadecimal into binary or decimal and then convert binary or decimal to octal.
COMPLEMENT
In a computer system, subtraction is not performed directly as arithmetic subtraction. It is performed by the technique called complement. It is the process of repeated addition.
 There are two types of complement: r`s complement and (r1)`s complement.
Where 'r' is the base of a number system.
In binary number system, there are two types of complement: 1`s complement and 2`s complement.
Similarly, decimal number system has 9`s and 10`s complement.

1`s Complement
1`s complement of a binary number is obtained by subtracting each bit by 1. We can get 1`s complement by simply replacing 1 by 0 and 0 by 1.
Example: 1`s complement of 1011 = 0100
Subtraction of binary numbers using 1`s complement
Steps are here as below:
 Make the both numbers having the same number of bits.
 Determine the 1`s complement of the number to be subtracted(subtrahend).
 Add the 1`s complement to the given number from which we subtract (minuend).
 If there exists` any additional bit (carry) in the result after addition, remove and add it to the result else (i.e. if there exists` no any carry)
 Determine the 1`s complement of the result and prefix by a negative sign to get the final result.
Example: Subtract 1110000 from 1100000

2`s complement
The 2`s complement of a binary number is obtained by adding binary 1 to the 1`s complement of the number.
Subtraction using 2`s complement:
Steps are here as below:
 Make the both numbers having the same number of bits.
 Determine the 2`s complement of the number to be subtracted (subtrahend).
 Add the 2`s complement to the given number from which we subtract (minuend).
 If there exits no carry determine the 2`s complement of the result and prefix by a negative sign to get a final result.
Example:Subtract 1110000 from 1100000

9`s Complement and 10`s Complement
The 9`s complement of decimal number can be obtained by subtracting each digit of the number from 9.
For example, the 9`s complement of 3 is 6 (93=6), and 234 is 765 (999234 =765).
The 10`s complement of decimal number can be obtained by adding 1 to the least significant digit of 9`s complement of that number. For example, 10`s complement of 3 is 7 (93=6+1=7), and 123 is 877.
Subtraction of decimal number using 9`s complement
Here are the steps are given below:
 Make the both numbers having the same number of digits.
 Determine the 9`s complement of the number from which we subtracted (subtrahend).
 Add the 9`s complement to the given number from which we subtract (minuend).
 If there exists` any additional digit (carry) in the result after addition, remove it and add it to the complement of the result and prefix by a negative sign to get the final result.
E.g. Subtract (123)_{10} From (345)_{10}
9`s complement of 123= (999 123) =876
Adding the 9`s complement with 345, i.e 345 + 876 = 1221
In the result, most significant digit 1 is the carryover. So add this carry over to remaining digits 221
i.e, 221 + 1 = 222
Hence, (222)_{10} is the required result after subtracting (123)_{10} from (345)_{10.}
Subtraction using 10`s complement:
Here are the steps are given below:
 Make the both numbers having same numbers of digits.
 Determine the 10`s complement of the number to be subtracted (subtrahend).
 Add the 10`s complement to the given number from which we subtract (minuend).
 If there exists`s any additional digit (carry) in the result after addition, remove it from the result and the remaining digits form the final result.
 If there exists` no any carry then determine the 10`s complement of the result and prefix by the negative sign to get the final result.
Example: Subtract (123)_{10} from (345)_{10}
10`s Complement of 123 = (999  123) = 876 + 1 = 877
Adding the 10`s complement with 345, i.e. 345 + 877 = 1222
In this result, most significant digit 1 is the carry over.So remove it to find the result.
Therefore, (222)_{10} is the required result.
Binary Mathematics
1. Binary Addition
Rule for binary addition0+0=01+0=10+1=11+1=10 (0 with carry over 1)  Example: Binary addition101101+101111000100 sum 
2. Binary Subtraction
Rule for binary subtraction 11=010=101=1 (with borrowing 1)00=0  Example: Binary addition101101 minuend10111 subtrahend10110 difference 
3. Binary Multiplication
Rule for binary multiplication1*1=11*0=00*1=00*0=0  Example: Binary multiplication1011 multiplicand *1011 multiplier10111011*0000** +1011***1111001 product 
4. Binary Division
Rule for binary division1/1=11/0=not defined0/1=00/0=not defined  Example: Binary DivisionDivide 101011 by 110110) 101011 (111 quotient 1101001 110111 1101 remainder 
Some Basic Terms Related with Number System
 MSB (Most Significant Bit)
The left most bit of a number is called MSB.
Example :
1010 = MBS
 LSB (Leas Significant Bit)
The right most bit of a number is called LSB.
Example:
1010 = LSB
BIT: Single binary number either 0 or 1
Nibble: Combination of 4 binary bits e.g. 1001
Byte: Combination of 8 binary bits e.g. 1001 0111
Logic Function and Boolean Algebra
Boolean algebra is the algebra of logic that deals with the study of binary variables and logical operations.
INTRODUCTION
 Boolean algebra is used to design and simplify circuits of electronic devices.
 Each input and output are thought as a member of the set {0, 1}.
 The basic elements of circuits are called gates. Each type of gate implements Boolean operation.
Boolean Algebra
Boolean algebra is the algebra of logic that deals with the study of binary variables and logical operations. It makes possible to transform logical statements into mathematical symbols and to calculate the truth or falsity of related statements by using rules. It is named after George Boole, a 19thcentury Mathematician and Philosopher, who was the first to try and to formalize what we call logic or reasoning.
This symbolic logic which was designed using variables and symbols is the language used to describe and manipulate logical statements and determine whether they are true or not. Through this, certain logical statements can be expressed in mathematical terms. For example, the symbols '•', '' and '+' represent 'and', 'not'and 'or'respectively. Aside from its many applications in the design of computers, it serves as the foundation of information theory.
Boolean variable
A boolean variable is the variables which have only two states i.e. true/ false or right/ wrong or on/off or 0/1. As a computer is a binary system, it operates on an electronic signal which has only 2 possible states.
The signal that does not change its state with time is called constant signal and its value always remains the same i.e. either 1 or 0 whereas a variable signal continuously changes its state according to the time. At some point, the value of the variable signal may be 1 and at some another point, it might be 0.Therefore, these variables which consist of only two values i.e. 1 and 0 are Boolean variables or logic variables. These variables are denoted by English capital letters like A, B, X, Y, etc.
Boolean algebra as Switching algebra:
Boolean Algebra which is also known as the Switching algebra consists of two elements (0 and 1) has two binary operators called OR and AND and another unary operator called NOT. The unary operator performs on a single operand whereas a binary operator requires more than one operand.
Followings are their symbols:
Operator 
Symbols 
OR 
~, +, V, U (Union) 
AND 
(.),^, Intersection 
NOT 
Ë…, (‘), (‾), Negation

Logic function (Boolean function)
Boolean function, commonly known as a logic function is an expression expressed algebraically with binary variables, logical operation symbols, parenthesis and equal sign. For a given value of the binary variables, the logic function can be either 0 or 1.
Example: Consider the logic function in algebraic expression:
F = X.Y.Z’+X.Y
Here, X, Y, Z are Boolean variables. The equation on the righthand side above is known as an expression. Each occurrence of a variable or its complement in an expression is called literal.So, there are altogether three variables i.e. X, Y and Z and five literals; X, Y, Z’, X, and Y.
Logical circuit
By using transistors or Integrated Circuits (IC) or (LSI), the high and lowlevel voltage electrical signals called logical elements are produced. These logical elements are essential for the composition of a circuit for a specific operation called logical circuit.
Basic Logical/ Boolean Operation:
Introduction:
An operator is a special symbol that indicates the operation to be carried out between two operands. An operation is an action to be carried out upon operands. There are 3 basic Boolean Operations: AND, OR and NOT operations.
 AND operation:
Known as logical multiplication, it is carried out by dot (.) operator or simply by AND. If the inputs are true, it generates true output. Otherwise, it generates false output. Its logical equation is written as C=A.B or C=A AND B. The truth table of AND operation is:
Inputs 
Output 

A 
B 
C=A.B 
False 
False 
False 
False 
True 
False 
False 
False 
False 
True 
True 
False 
 OR operation
Known as logical addition, it is carried out by plus (+) operator or simply by OR. If at least one input is true, it generates true output or else, it gives false output. The logical equation of OR operation is written as C=A+B or C=A OR B. The truth table of OR operation is given below:
Inputs 
Output 

A 
B 
C=A+B 
False 
False 
False 
False 
True 
True 
True 
False 
True 
True 
True 
True 
 NOT Operation
Also known as the logical compliment, it is carried out by prime (‘) operator or bar (‾). It generates the output opposite the input i.e. if the input is true, it generates false output and vice versa. It's logical equation can be written as C=A’. The truth table of OR operation is:
Inputs 
Output 
A 
C=A’ 
True 
False 
False 
True 
Truth table and Boolean expression:
The truth table is a table of all possible combinations of the variables showing the relation between the values that variables may take and the result of the operation. The table used to represent the Boolean expression of a logic gate function called a truth table. A truth table shows each possible input combination to the gate or circuit with the resultant output depending upon the combination of input. To represent a function in truth table, there should be the list of the combination of the binary variables. An expression that results in a Boolean value i.e. in a value of either true or false, which takes several forms.
The examples of truth table of different gates are as follows
Logic gates:
A logic gate is an electronic circuit that operates on one or more input signals to produce an output signal. A logic gate is also known building block of a digital circuit. Mostly, the logic gate consists of two inputs and one output. Gates produce the signals 1 or 0 if input requirements are satisfied. Digital computer uses different types of logical gates. Each gate has a specific function and graphical symbol. The function of the gate is expressed by means of an algebraic expression. The basic gates are described below:
 AND Gate:
The AND Gate contain two or more than to input values which produce only one output value. AND gate produces 1 output when all inputs are 1, otherwise the output will be 0. It can be explained with the help of two switches connected in series. In AND gate, current is flowing in the circuit only when both switches, A and B, are closed.
The switch contains two states which are ON or OFF. The ON means the logic 1 and the OFF means the logic 0. So, when both switches are ON, the output is 1 and when any of the switches are OFF, the output is 0.
The graphical symbol, logical circuit, algebraic expression and truth table of AND gate is shown below:
A  B  Y = A.B 
0  0  0 
0  1  0 
1  0  0 
1  1  1 
 OR Gate:
The OR Gate contains two or more than two input values which produce only one output value. OR gate produces 1 output, when one of the inputs is 1. If inputs are 0, then the output will be also 0. It can be explained by taking an example of two switches connected in parallel.
The graphical symbol, algebraic expression and truth table of OR gate is as shown below:
A  B  Y = A+B 
0  0  0 
0  1  1 
1  0  1 
1  1  1 
 NOT Gate:
The NOT Gate contains only one input value which produces only one output value. This gate is also known as an inverter. So, this circuit inverts the logical sense of a binary signal. It produces the complemented function. If the input is 1, then this gate will produce 0 as output and viceversa. The graphical symbol, algebraic expression and truth table of a NOT gate is given below.
A  A' 
0  1 
1  0 
 NAND Gate:
The NAND Gate contains two or more than two input values which produce only one output value. This gate is the combination of AND and NOT gates. This gate is a complement of AND function. This gate produces output 0, when all inputs are 1, otherwise, output will be 1.
The graphical symbol, algebraic expression and truth table of NAND gate is shown below:
A  B  A.B  y=(A.B)' 
0  0  0  1 
0  1  0  1 
1  0  0  1 
1  1  1  0 
 NOR Gate:
The NOR Gate contains two or more than two input values which produce only one output value. This gate is a combination of OR and NOT gate. This gate is the complement of the OR function. This gate produces 1 output, when all inputs are 0 otherwise output will 0.
The graphical symbol, algebraic expression and truth table of NOR gate are given below
A  B  A+B  Y = (A+B)' 
0  0  0  1 
0  1  1  0 
1  0  1  0 
1  1  1  0 
 Exclusive OR (XOR) Gate:
This gate contains two or more than two input values which produce only one output value. The graphical symbol of XOR gate is similar to OR gate except for the additional curve line on the input side. This gate produces 1 as output, if any input is 1 and 0 if both inputs are either 1 or 0, otherwise its output is 0.
The graphical symbol, algebraic expression and truth table of XOR gate is given below:
A  B  A'  B'  A'.B  A.B'  A'.B+A.B' 
0  0  1  1  0  0  0 
0  1  1  0  1  0  1 
1  0  0  1  0  1  1 
1  1  0  0  0  0  0 
 Exclusive NOR (XNOR) Gate:
This gate contains two or more than two input values which produce only one output value. The XNOR is the complement of the XOR, as indicated by the small circle in the graphical symbol. This gate produces 1 output, when all inputs are either 0 or 1, otherwise its output value is 0.
The graphical symbol, algebraic expression and truth table of XNOR gate is shown below
A  B  A'  B'  A.B  A'.B'  A.B+A'.B' 
0  0  1  1  0  1  1 
0  1  1  0  0  0  0 
1  0  0  1  0  0  0 
1  1  0  0  1  0  1 
LAW OF BOOLEAN ALGEBRA
As 0 and 1 are logic symbol that is used to represent a digital output or input that are used as constants permanently for “Open"
or “Closed” circuit rules. Boolean Algebra expression have been invented to help to reduce the number of logic gates that is used to perform a particular logic operation resulting a list of theorems or functions commonly knownas the "Laws of Boolean Algebra".
Boolean algebra was invented by world famous mathematician George Boole, in 1854. He published it in his book named "An Investigation of the Laws of Thought". There are theorems of these boolean that are used to make calculation fastest and easier ever than ever.
Boolean Algebra is Mathematics, that is used to analyze digital gates and circuits. We can use “Laws of Boolean” to reduce and simplify a complex Boolean expression to reduce the number of logic gates.
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)’ 
Associative Laws for Boolean Algebra
This law is for several variables, where the OR operation of the variable result is same though the grouping of the variables.This law is quite same in the case of AND operators.
 ( A . B ) . C = A . ( B . C )
 ( A + B ) + C = A + ( B + C)
Cumulative Law for Boolean Algebra
According to Cumulative Law, the order of OR operations and AND operations conducted on the variables make no difference.
 A. B = B. A
 A + B = B + A
Hence, the values fo A + B and B + A are both equal.
Distributive Laws for Boolean Algebra
This law can be composed of two operators: AND and OR.
 A . ( B + C) = (A . B) + (A . C)
Hence,the values of A . ( B + C ) and A. B + A . C are equal.
Identity Law for Boolean algebra
A term OR`ed with a "0" or AND with a "1" will always equal that term.
 A + 0 = A
 A.1 = A
Hence, the output will be same as the input.
Boolean Algebra Theorems
De Morgan’s Theorem:
De Morgan's theorem is associated with Boolean algebra, which was given by great logical and mathematician, De Morgan. So, it is called "De Morgan's theorem".
It consists of first and second theorem which are described below:

First Theorem:
The De Morgan’s first theorem states,
“The complement of the sum is equal to the product of complement of individual variable”
. Let P and Q be two Boolean variables then De Morgan’s theorem mathematically expressed as
(P + Q)’ = (P)’.(Q)’
Proof:
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 

Second Theorem:
De Morgan’s second theorem states,
” The complement of a product is equal to the sum of the complements of individual variable”.
Let P and Q be two Boolean variables then De Morgan’s theorem mathematically expressed as
(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 
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