DIGITAL ELCTRONICS/DIGITAL LOGIC DESIGN(DLD)

Combinational Logic Circuits: A combinational logic circuit is a type of digital circuit that combines multiple inputs to produce a single output based on a specific logic function. The output is determined solely by the current inputs, without any memory or feedback.

Different Types of Logic Gates: Logic gates are the basic building blocks of digital circuits. They perform logical operations on one or more binary inputs to produce a single binary output. The most common types of logic gates are:

• Buffer Gate
• NOT/INVERTER Gate
• AND Gate
• NAND Gate
• OR Gate
• NOR Gate
• XOR Gate
• XNOR Gate

Notes: Number of Outputs = 2^{Number of Inputs}

Buffer Gate: A buffer gate is a digital circuit that outputs the same logic level as its input. It does not change the signal, but it is used to strengthen the signal (provide more current or voltage drive) and isolate different parts of a circuit.
It does not produce high or low impedance states based on input.
Notes:

• A buffer always outputs HIGH if the input is HIGH, and LOW if the input is LOW.
• If you're thinking about high-impedance outputs, that applies to a tri-state buffer, which can have three states: HIGH, LOW, and high-impedance (Z).

Symbol:

Boolean Expression: A = F

Truth Table:

Input(A)	Output(F)
0	0
1	1

Logic Circuit Diagram:

Inverter/NOT Gate: A NOT gate, also known as an inverter, is a digital circuit that outputs the opposite logic level of its input. If the input is HIGH (1), the output is LOW (0), and vice versa.

Symbol:

Boolean Expression: F = A'

Truth Table:

Input(A)	Output(F)
0	1
1	0

Logic Circuit Diagram:

AND Gate: An AND gate is a digital circuit that outputs HIGH (1) only when all of its inputs are HIGH (1). If any input is LOW (0), the output is LOW (0).

Symbol:

Boolean Expression: F = A.B

Truth Table:

Input(A)	Input(B)	Output(F)
0	0	0
0	1	0
1	0	0
1	1	1

Logic Circuit Diagram:

NAND Gate: A NAND gate is a digital circuit that outputs LOW (0) only when all of its inputs are HIGH (1). If any input is LOW (0), the output is HIGH (1). It is the negation of the AND gate.

Symbol:

Boolean Expression: F = (A.B)'

Truth Table:

Input(A)	Input(B)	Output(F)
0	0	1
0	1	1
1	0	1
1	1	0

Logic Circuit Diagram:

OR Gate: An OR gate is a digital circuit that outputs HIGH (1) if at least one of its inputs is HIGH (1). If all inputs are LOW (0), the output is LOW (0).

Symbol:

Boolean Expression: F = A + B

Truth Table:

Input(A)	Input(B)	Output(F)
0	0	0
0	1	1
1	0	1
1	1	1

Logic Circuit Diagram:

NOR Gate: A NOR gate is a digital circuit that outputs LOW (0) if at least one of its inputs is HIGH (1). If all inputs are LOW (0), the output is HIGH (1). It is the negation of the OR gate.

Symbol:

Boolean Expression: F = (A + B)'

Truth Table:

Input(A)	Input(B)	Output(F)
0	0	1
0	1	0
1	0	0
1	1	0

Logic Circuit Diagram:

XOR Gate: An XOR (exclusive OR) gate is a digital circuit that outputs HIGH (1) if an odd number of its inputs are HIGH (1). If both inputs are the same, the output is LOW (0).

Symbol:

Boolean Expression:F = A ⊕ B = (A.B') + (A'.B)

Truth Table:

Input(A)	Input(B)	Output(F)
0	0	0
0	1	1
1	0	1
1	1	0

Logic Circuit Diagram:

XNOR Gate: An XNOR (exclusive NOR) gate is a digital circuit that outputs HIGH (1) if an even number of its inputs are HIGH (1). If both inputs are the same, the output is HIGH (1); otherwise, it is LOW (0).

Symbol:

Boolean Expression: F = A ⊙ B= (A ⊕ B)' = (A.B) + (A'.B')

Truth Table:

Input(A)	Input(B)	Output(F)
0	0	1
0	1	0
1	0	0
1	1	1

Logic Circuit Diagram:

Postulates of Boolean Algebra: The postulates of Boolean algebra are a set of fundamental rules that govern the behavior of Boolean variables and operations. These postulates are used to simplify and manipulate Boolean expressions. The main postulates include:

De Morgan's Laws: De Morgan's laws are a pair of transformation rules that are used to simplify complex logical expressions. They state that:

Universal Gates: Universal gates are logic gates that can be used to create any other type of logic gate. The two main types of universal gates are NAND and NOR gates.

INVERTER/NOT GATE USING NAND AND NOR GATES

NOT GATE USING NAND GATE: To create a NOT gate using a NAND gate, connect both inputs of the NAND gate to the same input signal. The output will be the inverted signal.

Symbol:

Boolean Expression: F = (A.A)'= A'

Truth Table:

Input(A)	Output(F)
0	1
1	0

NOT GATE USING NOR GATE: To create a NOT gate using a NOR gate, connect both inputs of the NOR gate to the same input signal. The output will be the inverted signal.

Symbol:

Boolean Expression: F = (A + A)'= A'

Truth Table:

Input(A)	Output(F)
0	1
1	0

AND GATE USING NAND AND NOR GATES

AND GATE USING NAND GATE: To create an AND gate using NAND gates, connect the output of a NAND gate to the input of another NAND gate, with both inputs of the second NAND gate connected to the output of the first NAND gate.

Symbol:

Boolean Expression: F = ((A.B)'.(A.B)')' = ((A.B)')' = A.B

Truth Table:

Input(A)	Input(B)	Output(F)
0	0	0
0	1	0
1	0	0
1	1	1

AND GATE USING NOR GATE: To create an AND gate using NOR gates, connect the output of two NOR gates to the input of another NOR gate with each NOR gate receiving one input signal.

Symbol:

Boolean Expression: F = ((A + A)' + (B + B)')' = (A' + B')' = ((A.B)')' = A.B

Truth Table:

Input(A)	Input(B)	Output(F)
0	0	0
0	1	0
1	0	0
1	1	1

OR GATE USING NAND AND NOR GATES

OR GATE USING NAND GATE: To create an OR gate using NAND gates, connect the output of two NAND gates to the input of a third NAND gate, with each NAND gate receiving one input signal.

Symbol:

Boolean Expression: F = ((A.A)'.(B.B)')' = (A'.B')' = (A')' + (B')' = A + B

Truth Table:

Input(A)	Input(B)	Output(F)
0	0	0
0	1	1
1	0	1
1	1	1

OR GATE USING NOR GATE: To create an OR gate using NOR gates, connect the output of a NOR gate to the input of another NOR gate, with each NOR gate receiving one input signal.

Symbol:

Boolean Expression: F = ((A + B)' + (A + B)')' = ((A + B)')' + ((A + B)')' = (A + B) + (A + B) = A + B

Truth Table:

Input(A)	Input(B)	Output(F)
0	0	0
0	1	1
1	0	1

NAND GATE USING NAND AND NOR GATES

NAND GATE USING NAND GATE:

Symbol:

Boolean Expression: F = (A.B)'

Truth Table:

Input(A)	Input(B)	Output(F)
0	0	1
0	1	1
1	0	1
1	1	0

NAND GATE USING NOR GATE: To create a NAND gate using NOR gates, connect the output of two NOR gates to the inputs of a third NOR gate, with each NOR gate receiving one input. Then connect the output of the third NOR gate to the inputs of the final NOR gate receiving one input signal.

Symbol:

Boolean Expression: F = (((A + A)' + (B + B)')' + ((A + A)' + (B + B)')')' = ((A' + B')' + (A' + B')')' = ((A')'.(B')'+ (A')'.(B')')' = ((A.B) + (A.B))' = (A.B)'

Truth Table:

Input(A)	Input(B)	Output(F)
0	0	1
0	1	1
1	0	1
1	1	0

NOR GATE USING NAND AND NOR GATES

NOR GATE USING NAND GATE: To create a NOR gate using NAND gates, connect the output of two NAND gates to the inputs of a third NAND gate, with each NAND gate receiving one input signal. Then connect the output of the third NAND gate to the inputs of the final NAND gate receiving one input signal.

Symbol:

Boolean Expression: F = (((A.A)'.(B.B)')'.((A.A)'.(B.B)')')' = ((A'.B')'.(A'.B')')' = (((A')' + (B')').((A')' + (B')'))' = ((A + B).(A + B))' = (A + B)'

Truth Table:

Input(A)	Input(B)	Output(F)
0	0	1
0	1	0
1	0	0
1	1	0

NOR GATE USING NOR GATE:

Symbol:

Boolean Expression: F = (A + B)'

Truth Table:

Input(A)	Input(B)	Output(F)
0	0	1
0	1	0
1	0	0
1	1	0

XOR GATE USING NAND AND NOR GATES

XOR GATE USING NAND GATE: To create an XOR gate using NAND gates, connect the output of A NANDs and the input of B to a NAND gate. Also connect A input and the output of B NANDs to another NAND gate. Finally, connect the both NANDs output to another NAND gate.

Symbol:

Boolean Expression: F = ((A'.B)' . (A.B')')' = ((A'.B)')' + ((A.B')')' = A'.B + A.B' = A ⊕ B

Truth Table:

Input(A)	Input(B)	Output(F)
0	0	0
0	1	1
1	0	1
1	1	0

XOR GATE USING NOR GATE: To create an XOR gate using NOR gates, connect the input A and B to the input of a NOR gate. Also connect the input A and B NORs output to another NOR gate. Now connect both NOR gates outputs to another NOR gate inputs.

Symbol:

Boolean Expression: F = ((A' + B')' + (A + B)')' = (((A' + B')')'.((A + B)')') = (A' + B').(A + B) = A'A + A'B + AB' + BB' = A'B + AB' = A ⊕ B

Truth Table:

Input(A)	Input(B)	Output(F)
0	0	0
0	1	1
1	0	1
1	1	1

XNOR GATE USING NAND AND NOR GATES

XNOR GATE USING NAND GATE: To create an XNOR gate using NAND gates, connect the input of A and B to the input of a NAND gate. Also connect the outputs of A and B NANDs to another NAND gate. Finally, connect the outputs of both NAND gates to another NAND gate.

Symbol:

Boolean Expression: F = ((A.B)'.(A'.B')')' = (((A.B) + (A'.B'))')' = A.B + A'.B' = A ⊙ B [= (A ⊕ B)']

Truth Table:

Input(A)	Input(B)	Output(F)
0	0	1
0	1	0
1	0	0
1	1	0

XNOR GATE USING NOR GATE: To create an XNOR gate using NOR gates, connect the input of A and B to the input of a NOR gate. Then connect the outputs of A and B NORs to another two NOR gates input and each NOR gate another input should be input A for one and input B for the other. Finally, connect the outputs of both NOR gates to another NOR gate.

Symbol:

Boolean Expression: F = ((A + (A + B)')' + (B + (A + B)')')' = (A'.((A + B)')' + B'.((A + B)')')' = (A'.(A + B) + B'.(A + B))'' = (A'.A + A'.B + A.B' + B.B')' = (A'.B + A.B')' = (A ⊕ B)' = A ⊙ B

Truth Table:

Input(A)	Input(B)	Output(F)
0	0	1
0	1	0
1	0	0
1	1	1