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Chapter 14: Problem 3

How could you use inverters and NAND gates to create an AND gate, a logic element that has two inputs and one output, with the output being 1 only if both of the inputs are also 1s?

Short Answer

Expert verified
Use two NAND gates: the first acts as a NAND gate, and the second as an inverter to create an AND gate.

Step by step solution

01

Understanding NAND Gate

A NAND gate is a digital logic gate that outputs 0 only if all its inputs are 1. Otherwise, it outputs 1. The truth table for a NAND gate with inputs A and B and output Q is: \[\begin{array}{cc|c}A & B & Q \hline0 & 0 & 1 \0 & 1 & 1 \1 & 0 & 1 \1 & 1 & 0 \end{array}\]
02

Create Inverter using NAND Gate

An inverter (NOT gate) can be created using a single NAND gate by connecting both inputs of the NAND gate to the same input signal. If input is A, then the output will be \( \overline{A} \).
03

Combine NAND Gates to Form AND Gate

To create an AND gate using only NAND gates and inverters, follow these steps:1. Take two inputs, A and B.2. Connect A and B to the inputs of the first NAND gate. Let's call the output of this gate Y: \( Y = \overline{A \wedge B} \).3. Use an inverter (created from a NAND gate) to invert the output Y. The final output will be \( \overline{Y} \). Since \( Y = \overline{A \wedge B} \), \( \overline{Y} \) will be \( A \wedge B \).
04

Verify the Construction

Verify the truth table of the constructed AND gate: \[\begin{array}{cc|c}A & B & Q \hline0 & 0 & 0 \0 & 1 & 0 \1 & 0 & 0 \1 & 1 & 1 \end{array}\]This truth table matches the expected behavior of an AND gate.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

NAND Gate
A NAND gate is one of the fundamental building blocks in digital logic design. It's a combination of an AND gate followed by a NOT gate. The NAND gate only outputs a 0 if all its inputs are 1. Otherwise, the output is always 1. This property makes the NAND gate extremely versatile, as you can create other types of gates from it.
Here's the truth table for a NAND gate:
  • 0 + 0 = 1
  • 0 + 1 = 1
  • 1 + 0 = 1
  • 1 + 1 = 0
By understanding how a NAND gate works, you can create more complex circuits, such as an AND gate from just NAND gates and inverters.
Inverter
An inverter, also known as a NOT gate, is a logic gate that outputs the opposite value of its input. In simple terms, it turns a 0 into a 1 and a 1 into a 0. You can create an inverter using a NAND gate.
Here's how:
  • Take a single input signal, let's call it A.
  • Connect both inputs of a NAND gate to this input signal A.
  • The output of the NAND gate will then be \( \overline{A} \), which is the inverted value of A.
This trick of creating an inverter is handy when you want to build more complex logic gates, such as an AND gate, from NAND gates.
AND Gate
An AND gate is another essential element in digital logic design. Its function is to output a 1 only when all its inputs are also 1. If any input is 0, the output will be 0.
    Here's the truth table for an AND gate:
  • 0 and 0 = 0
  • 0 and 1 = 0
  • 1 and 0 = 0
  • 1 and 1 = 1
You can create an AND gate using a combination of NAND gates and inverters:
  • Take two inputs, A and B.
  • Connect them to the inputs of the first NAND gate. Call the output of this gate Y: \( Y = \overline{A \wedge B} \).
  • Use an inverter (created from a NAND gate) to invert the output Y. The final output will be \( \overline{Y}\). Since \( Y = \overline{A \wedge B}\), \( \overline{Y}\) will be equivalent to \( A \wedge B \).
By verifying the construction with a truth table, you can confirm that the combination of NAND gates and an inverter successfully mimics an AND gate.

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