Chapter 2- Boolean Logic
Logic Gates
-Digital Systems are said to be built using logic gates.
-A logic gate is an electronic circuit or logic circuit which can take one or more than one input to get only one output.
Types of logic Gates:-
1. AND Gate
2. OR Gate
3. NOT Gate
4. NOR Gate
5. NAND Gate
6. XOR Gate
7. XNOR Gate
1) AND Gate:
The AND gate contains two or more than two input values which produce only two one output value. AND gate produces 1 output when all inputs are 1, otherwise the output will be 0.
A | B | Y=A.B |
0 | 0 | 0 |
0 | 1 | 0 |
1 | 0 | 0 |
1 | 1 | 1 |
Logic Symbol
Boolean expression
Y=A.B
Venn Diagram
2) OR Gate:
The of Gate contains two or more input with single output.It produce 1 output, when one of the input is 1, & 0 when all inputs are 0.
3) NOT Gate
The NOT gate contains only one input value which produces only one output value. This gate is also known as an inverter which inverts the logical sense of binary system signal.
4) NAND
The NAND gate contains two or more than two input values which produce only one output value.This gate produces output 0, when all inputs are 1, otherwise the output will be 1.
5) NOR Gate
This gate is a combination of OR and Not gate or it is a complement of OR gate which produces
1 output when input is zero, otherwise the output will be 0.
6) Exclusive OR (XOR) Gate
This gate contains two or more than two input values which produces only one output value. The logical 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.
7) Exclusive NOR (XNOR) Gate
The XNOR is the complement of XOR gate which produce 1 output when all input are same, otherwise 0.
Multiple Choice Questions:
a) The logic gate which produces result TRUE only when all inputs are false is
i. AND
ii. OR
iii. NAND
iv. NOR
b) Which of the following statement is not correct?
i. A + A' = 1
ii. A + A = A
iii. A .1 = A
iv. (A + B)' = A' + B'
c) Which of the following logic gates is known as universal gate?
i. AND
ii. OR
iii. NOR
iv. XOR
d) Which of the following logic gate accepts exact two inputs?
i. NAND
ii. NOT
iii. AND
iv. XOR
e) The boolean expression C + (BC)' is equivalent to
i. 0
ii. C
iii. B
iv. 1
f)Which of the following is considered as idempotent law?
i. A + A = A
ii. A + 1 = 1
iii. A + A' = 1
iv. A.A = 2A
g) The boolean expression A. (A + B) is equivalent to
i. 0
ii. B
iii. 1
iv. A
h) Which of the following logic gates can be designed to develop all other logic gate
i. NAND
ii. XOR
iii. OR
iv. XNOR
i) The result produced by ...gate will be False(0) if both inputs are differ otherwise True(1).
i. XOR
ii. XNOR
iii. OR
iv. NOT
j) NOR gate is formed by the combination of OR gate with..........gate
i. NAND
ii. OR
iii. NOT
iv. AND
2. Short Answer Questions:
i.What is universal gate? Why NAND and NOR are called universal gates?
A universal gate is a logic gate which can be used to implement all basic logic functions (AND, OR, NOT) without using any other type of gate.
NAND and NOR are called universal gates because:
By combining NAND (or NOR) gates in suitable arrangements, we can construct the three basic gates (NOT, AND, OR).
Since all digital circuits are built from these basic gates, NAND and NOR alone are sufficient to design any logic circuit.
//LOGIC SYMBOLS//
ii. State and prove De Morgan's theorem for two input variables.
ii. De Morgan’s Theorem (for two variables)
The theorem states:
1. (A + B)’ = A’ · B’
2. (A · B)’ = A’ + B’
//TRUTH TABLE AND LOGIC SYMBOLS//
iii.Write the truth table and draw the logical diagram for the following boolean expression
a. F = C + (BC)'
b. F = AB + A' C + B C'
C. F = (A+B) + (A + C)
d. F=(A.B.C)+A. (B+ C)'
e.F=AB+ BC(B + C)
3.Simplify the following boolean expression by using laws of boolean algebra.
a) AB + BC(B + C)
BC(B + C) = BCB + BCC = BC + BC = BC` (Distributive, Idempotent)
2. `F = AB + BC = B(A + C)` (Factoring)
⇒ F = B(A + C)
b) C + (BC)'
1. `(BC)' = B' + C'` (De Morgan)
2. `C + (B' + C') = (C + B') + (C + C') = (C + B') + 1 = 1` (Complement, Domination)
⇒ 1
c) (A + B)(A + C)
1. `= A(A + C) + B(A + C)` (Distributive)
2. `= A + AB + BC` (Identity, Distributive)
3. `A + AB = A` (Absorption)
⇒ A + BC
d) AB + A(B + C) + B(B + C)
1. `A(B + C) = AB + AC`, `B(B + C) = B + BC` (Distributive, Idempotent)
2. `F = AB + AB + AC + B + BC = AB + AC + B + BC`
3. `B + AB = B`, `B + BC = B` (Absorption)
⇒ B + AC
e) AB' + A(B + C)' + B(B + C)'
1. `(B + C)' = B'C'` (De Morgan)
2. `A(B + C)' = AB'C'`, `B(B + C)' = BB'C' = 0` (Annihilation)
3. `F = AB' + AB'C' = AB'(1 + C') = AB'` (Absorption, Identity)
⇒ AB'
f) [AB'(C + BD) + A'B']·C
1. `AB'(C + BD) = AB'C + AB'BD = AB'C + 0 = AB'C` (Annihilation $B'B=0$)
2. Inside bracket: `AB'C + A'B'`
3. Multiply by `C`: `(AB'C + A'B')C = AB'C + A'B'C = B'C(A + A') = B'C` (Complement, Factor)
⇒ B'C
g) A'BC + AB'C' + A'B'C' + AB'C + ABC
1. Group: `(A'BC + ABC) = BC(A' + A) = BC` (Complement)
2. `(AB'C' + AB'C) = AB'(C' + C) = AB'` (Complement)
3. Remaining term: `A'B'C'`
4. Combine `AB' + A'B'C' = B'(A + A'C') = B'(A + C') = AB' + B'C'` (Consensus/Absorption)
5. With step (1): `F = BC + AB' + B'C'`
⇒ AB' + BC + B'C' (minimal sum)
4. State Duality principle with examples.
The duality principle states that every algebraic expression in Boolean algebra remains valid if we interchange AND (·) with OR (+) and interchange 0 with 1.
In other words, for every Boolean identity, there is a dual identity.
Examples:
1. Identity Law
Original: A + 0 = A
Dual: A . 1 = A
2. Null Law
Original: A + 1 = 1
Dual: A . 0 = 0
3. Idempotent Law
Original: A + A = A
Dual: A . A = A
5. Write short notes on:
a) Complement Law:
The complement of a Boolean variable is its inverse.
It is denoted by A'
Laws:
1. A + A' = 1 (A OR NOT A = 1)
2. A . A' = 0 (A AND NOT A = 0)
b) Distributive Law:
Similar to arithmetic distributive law but for Boolean algebra:
1. A . (B + C) = (A .B) + (A . C)
2. A + (B . C) = (A + B) . (A + C)
c) Boolean Function:
A Boolean function is a function that takes one or more binary inputs (0 or 1) and produces a binary output (0 or 1).
Example: F(A,B) = A . B + A'
Boolean functions are the foundation of digital circuits and can be implemented using logic gates.
6.Long Answer Questions:
i. What is logic gate? Explain any four types of gates with truth table and logical diagram.
ii.State and prove De Morgan's theorem for three input variables.
iii. State and prove any four laws of Boolean algebra by using truth table.
iv. What is boolean algebra? Explain AND, XOR, NOR and NOT gate with truth table, logical symbol and Venn diagram.