This slide contains the class notes of Class XI.

1. Chapter 4. Logic
Function and
Boolean Algebra

2. Definition:
- Is the algebra of two valued system of logic that determines the logical
prepositions in terms of True or False.
- The logic function and Boolean algebra both are integrated in one to design a logic
circuit of computer or any electronic circuits.

3. Example:
● Proposition : Any True or False declarative sentence is termed as proposition or
statement.
- George Boole was mathematician : True statement
- The Sun rises in the west : False statement
● Negation: When a statement is presented by its contradiction, then that
statement is known as negation statement.
- let statement p = the king is brave then its negation is written as ~p = the king is
not brave.
- if p is true then ~p is false and if p is false then ~p is true.

4. Boolean function:
- Is an expression formed with binary variables, the two binary operators OR and
AND, the unary operator NOT, parenthesis and equal sign.
- Example:
Boolean function F= ( abc ) is equal to 1
If a = 1 AND b = 1 AND c = 1, otherwise F = 0
- Boolean function is represented as an algebraic expression as well as truth table.
A B A.B
0 0 0
0 1 0
1 0 0
1 1 1

5. Boolean operator:
- Is a symbol that performs and indicates any operation between two or more
operands.4
- There basic operators in boolean function:
a. AND operator ( Product operator)
b. OR operator ( Addition operator )
c. NOT operator ( reverse operator )

6. Logic Gate:
- Is an electronic circuit to receive more than one input and deliver single output.
- Gates are often called as logic circuit because they can be analyzed with boolean
algebra.
- The computer system is a set of gates.
- ALU is responsible for mathematical and logical processing of data.
- Three basic gates in digital computer:
a. AND gate
b. OR gate
c. NOT gate

7. AND gate:
- Is a type of logic gate which produces high (1) or True output when all inputs are
high(1), otherwise the output will be low(0) of false.
- Algebraic Expression : X = A . B

8. Truth table of AND gate:
Input Output
A B A.B
0 0 0
0 1 0
1 0 0
1 1 1

9. OR gate:
- Is a type of logic gate, which produces high (1) or true output when any one of the
input is high (1) or true.
- If all the input are low ( 0 )or false then the output will be low (0).
- Algebraic expression : X = A + B

10. Truth table of OR gate:
Input Output
A B A + B
0 0 0
0 1 1
1 0 1
1 1 1

11. NOT gate:
- Is a type of logic gate in which the output will be the complement or just reverse of
input.
- If the input will be low (0) or false then the output will be high (1) or true and vice
versa.
- Algebraic expression : X = A ‘ or A
A

12. Truth table of NOT gate:
Input Output
0 1
1 0

13. Comparison of AND, OR and NOT gate:
AND gate OR gate NOT gate
- Receives more than
one input and
produces only one
output
- Receives more than
one input and
produces only one
output
- Receives only one
input and gives only
one output
- If all signal are high,
the output will be high
- If anyone input signal
is high, the output
signal becomes high.
- It inverts high input
into low and low into
high, so called as
inverter.

14. NAND gate:
- Is the combination of AND and NOT gate.
- Is complement of AND gate.
- Algebraic expression : X = ( A . B) ’

15. Truth table of NAND gate:
Input
X = A . B X = ( A . B ) ‘
A B
0 0 0 1
0 1 0 1
1 0 0 1
1 1 1 0

16. NOR gate:
- Is the combination of OR and NOT gate.
- Is the complement of OR gate.
- Produces high (1) output when all inputs are low (0) otherwise, the output will be
low (0).
- Algebraic expression : X = ( A+B ) ‘

17. Truth table of NOR gate:
Input
X = A + B X = ( A + B ) ‘
A B
0 0 0 1
0 1 1 0
1 0 1 0
1 1 1 0

18. X-OR gate:
- Is exclusive OR gate which produces low (0) output when all the inputs are same
otherwise, the output will be high (1).
- Algebraic expression : X = A ’ . B + A . B ’

19. Truth table of X-OR gate:
Input Output
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

20. X-NOR gate:
- Is exclusive NOR gate.
- Is just the complement of X-NOR gate which produces high (1) output when all the
inputs are either low (0) or high (1).
- Algebraic expression : X = A . B + A ‘ . B ’

21. Truth table of X-NOR gate:
Input Output
A B A’ B’ A’.B’ A.B A.B +
A’.B’
0 0 1 1 1 0 1
0 1 1 0 0 0 0
1 0 0 1 0 0 0
1 1 0 0 0 1 1

22. NAND and NOR gate as universal gate:
- Because these gates are efficient to implement any boolean function.
- The combination of NAND gate can be used to perform AND and NOT operation.
- The combination of NOR gate can be used to perform OR and NOT operation.

23. De-Morgan’s theorem:
● First theorem : “ The complement of sum equals to the product of the
complement.”
● Mathematically : ( A + B ) ‘ = A’ . B’
Input Ouput 1 Output 2
A B A + B ( A +B )’ A’ B’ A’ . B’
0 0 0 1 1 1 1
0 1 1 0 1 0 0
1 0 1 0 0 1 0
1 1 1 0 0 0 0

24. ● Second theorem : “ The complement of a product is equal to the sum of the
complement. ”
● Mathematically: ( A . B ) ‘ = A’ + B’
Input Ouput 1 Output 2
A B A . B ( A . B )’ A’ B’ A’ + B’
0 0 0 1 1 1 1
0 1 1 1 1 0 1
1 0 1 1 0 1 1
1 1 1 0 0 0 0

25. Different laws of Boolean Algebra:
● Commutative law : i) A +B = B + A ii) A . B = B . A
● Distributive law : A . ( B + C ) = A . B + A . C
● Complement law : i) ( A’ ) ‘ = A ii) A . A’ = A
● Identity law : i) A . 0 = 0 ii) A + 0 = A iii) A . 1 = A
iv) A +1 =1
● Associative law : i) A + ( B + C ) = ( A + B) + C ii) A . ( B . C) = (A .
B) . C