Why NAND and NOR gate are called universal?
1 Answer
NAND and NOR gates are often called **universal gates** because they are functionally complete, meaning that they can be used to create any other logic gate (such as AND, OR, NOT, XOR, etc.) through various combinations. This is a fundamental concept in digital logic design, which makes NAND and NOR gates extremely versatile in building complex circuits.
Here’s why they are considered universal:
### **1. NAND Gate as a Universal Gate**
A **NAND** (NOT AND) gate outputs false (0) only when both inputs are true (1), otherwise, it outputs true (1). The truth table for a NAND gate is as follows:
| Input A | Input B | Output (A NAND B) |
|---------|---------|-------------------|
| 0 | 0 | 1 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 0 |
To see why NAND is universal, let's look at how you can use a NAND gate to create other basic gates:
- **NOT Gate:** You can create a NOT gate by connecting both inputs of the NAND gate to the same value. For instance, for an input \(A\), the output of \(A \text{ NAND } A\) is \( \overline{A} \) (the negation of \(A\)).
- **AND Gate:** By taking the output of a NAND gate and applying a NOT gate to it (which can also be made using a NAND gate), we can create an AND gate. That is, \( A \text{ AND } B = \overline{(A \text{ NAND } B)} \).
- **OR Gate:** An OR gate can be constructed by applying the NAND gate in the following way: \( A \text{ OR } B = \overline{(\overline{A} \text{ NAND } \overline{B})} \).
Thus, a single NAND gate can be used to simulate all the basic logic gates, making it universal.
### **2. NOR Gate as a Universal Gate**
A **NOR** (NOT OR) gate outputs true (1) only when both inputs are false (0), otherwise, it outputs false (0). The truth table for a NOR gate is as follows:
| Input A | Input B | Output (A NOR B) |
|---------|---------|------------------|
| 0 | 0 | 1 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 0 |
Similarly to NAND, the NOR gate can be used to create other basic gates:
- **NOT Gate:** A NOT gate can be made by connecting both inputs of the NOR gate to the same value. For instance, for an input \( A \), the output of \( A \text{ NOR } A \) is \( \overline{A} \).
- **OR Gate:** By applying a NOT gate (using NOR) to the output of a NOR gate, we can construct an OR gate. That is, \( A \text{ OR } B = \overline{(A \text{ NOR } B)} \).
- **AND Gate:** An AND gate can be constructed by using a NOR gate and then applying NOR to the inverted inputs. For instance, \( A \text{ AND } B = \overline{(\overline{A} \text{ NOR } \overline{B})} \).
Thus, a NOR gate, just like the NAND gate, can be used to create all the basic logic gates.
### **Summary**
The key idea behind the term "universal" is that both NAND and NOR gates are capable of implementing any Boolean function. This versatility makes them fundamental building blocks in digital circuit design. By combining just a few of these gates in various configurations, any other logic operation can be performed. Because of this, NAND and NOR gates are often favored in the design of integrated circuits and digital systems, where simplicity and efficiency are critical.
Here’s why they are considered universal:
### **1. NAND Gate as a Universal Gate**
A **NAND** (NOT AND) gate outputs false (0) only when both inputs are true (1), otherwise, it outputs true (1). The truth table for a NAND gate is as follows:
| Input A | Input B | Output (A NAND B) |
|---------|---------|-------------------|
| 0 | 0 | 1 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 0 |
To see why NAND is universal, let's look at how you can use a NAND gate to create other basic gates:
- **NOT Gate:** You can create a NOT gate by connecting both inputs of the NAND gate to the same value. For instance, for an input \(A\), the output of \(A \text{ NAND } A\) is \( \overline{A} \) (the negation of \(A\)).
- **AND Gate:** By taking the output of a NAND gate and applying a NOT gate to it (which can also be made using a NAND gate), we can create an AND gate. That is, \( A \text{ AND } B = \overline{(A \text{ NAND } B)} \).
- **OR Gate:** An OR gate can be constructed by applying the NAND gate in the following way: \( A \text{ OR } B = \overline{(\overline{A} \text{ NAND } \overline{B})} \).
Thus, a single NAND gate can be used to simulate all the basic logic gates, making it universal.
### **2. NOR Gate as a Universal Gate**
A **NOR** (NOT OR) gate outputs true (1) only when both inputs are false (0), otherwise, it outputs false (0). The truth table for a NOR gate is as follows:
| Input A | Input B | Output (A NOR B) |
|---------|---------|------------------|
| 0 | 0 | 1 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 0 |
Similarly to NAND, the NOR gate can be used to create other basic gates:
- **NOT Gate:** A NOT gate can be made by connecting both inputs of the NOR gate to the same value. For instance, for an input \( A \), the output of \( A \text{ NOR } A \) is \( \overline{A} \).
- **OR Gate:** By applying a NOT gate (using NOR) to the output of a NOR gate, we can construct an OR gate. That is, \( A \text{ OR } B = \overline{(A \text{ NOR } B)} \).
- **AND Gate:** An AND gate can be constructed by using a NOR gate and then applying NOR to the inverted inputs. For instance, \( A \text{ AND } B = \overline{(\overline{A} \text{ NOR } \overline{B})} \).
Thus, a NOR gate, just like the NAND gate, can be used to create all the basic logic gates.
### **Summary**
The key idea behind the term "universal" is that both NAND and NOR gates are capable of implementing any Boolean function. This versatility makes them fundamental building blocks in digital circuit design. By combining just a few of these gates in various configurations, any other logic operation can be performed. Because of this, NAND and NOR gates are often favored in the design of integrated circuits and digital systems, where simplicity and efficiency are critical.