Design some simple digital circuits based on Boolean expressions. Draw circuits that implement the following Boolean expressions using some combination of AND, OR, NOT, NAND, and NOR gates.

- NOT X OR NOT Y OR Z
- X OR Y OR NOT Z
- NOT X AND Y OR NOT Y AND Z
Write a two to three page paper discussing your design. Please include the following:
1. Diagrams of your logic circuits.
2. Truth tables that indicate the output (a 0 or a 1) for each possible combination of values of X, Y, and Z.

This is one of those where you need to post your suggestions and we will comment. From a personal point of view I like to convert the words into a Boolean symbols expression before I do anything else. You may have done the reverse to post it here, of course.

If it is useful there are truth tables for the various gates here:

http://en.wikipedia.org/wiki/Logic_gate

It is not clear in your question if you can use multiple input gates, such as 3-input OR Gates?

To design digital circuits based on Boolean expressions, we'll start by analyzing the given expressions and then use different logic gates (AND, OR, NOT, NAND, and NOR) to implement them. Let's go through each expression one by one.

Expression 1: NOT X OR NOT Y OR Z
To design a circuit for this expression, we can use three gates: one NOT gate and two OR gates.

1. Assign inputs:
- X, Y, and Z are the inputs for the expression.

2. Implement the circuit:
- Connect X to the input of a NOT gate.
- Connect Y to the input of another NOT gate.
- Connect Z directly to an OR gate.
- Connect the output of the first NOT gate and the second NOT gate to the inputs of the second OR gate.
- The output of the second OR gate will be the result of the expression.

3. Diagram:
- Here's a diagram of the circuit for the expression "NOT X OR NOT Y OR Z":
- (NOT X) OR (NOT Y) OR Z

![Expression 1 Circuit Diagram](https://i.imgur.com/jtoqGBu.png)

4. Truth table:
- To determine the output for each possible combination of X, Y, and Z, we can create a truth table.
- The truth table for "NOT X OR NOT Y OR Z" will have 8 rows, where each row represents a unique combination of inputs (X, Y, Z) and shows the corresponding output.

| X | Y | Z | NOT X OR NOT Y OR Z |
|---|---|---|---------------------|
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 1 |
| 0 | 1 | 0 | 1 |
| 0 | 1 | 1 | 1 |
| 1 | 0 | 0 | 0 |
| 1 | 0 | 1 | 1 |
| 1 | 1 | 0 | 0 |
| 1 | 1 | 1 | 1 |

Expression 2: X OR Y OR NOT Z
For this expression, we'll again use three gates: two OR gates and one NOT gate.

1. Assign inputs:
- X, Y, and Z are the inputs for the expression.

2. Implement the circuit:
- Connect Z to the input of a NOT gate.
- Connect X and Y directly to two separate inputs of an OR gate.
- Connect the output of the NOT gate and the OR gate to the inputs of another OR gate.
- The output of the second OR gate will be the result of the expression.

3. Diagram:
- Here's a diagram of the circuit for the expression "X OR Y OR NOT Z":
- X OR Y OR (NOT Z)

![Expression 2 Circuit Diagram](https://i.imgur.com/vWiEaLC.png)

4. Truth table:
- The truth table for "X OR Y OR NOT Z" will show the output for each combination of inputs (X, Y, Z).

| X | Y | Z | X OR Y OR NOT Z |
|---|---|---|-----------------|
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 1 |
| 0 | 1 | 0 | 1 |
| 0 | 1 | 1 | 1 |
| 1 | 0 | 0 | 1 |
| 1 | 0 | 1 | 1 |
| 1 | 1 | 0 | 1 |
| 1 | 1 | 1 | 0 |

Expression 3: NOT X AND Y OR NOT Y AND Z
Here, we'll need a combination of four gates: two AND gates, one OR gate, and one NOT gate.

1. Assign inputs:
- X, Y, and Z are the inputs for the expression.

2. Implement the circuit:
- Connect X to the input of a NOT gate.
- Connect X and Y to two separate inputs of an AND gate.
- Connect Y to the input of another NOT gate.
- Connect Y and Z to two separate inputs of another AND gate.
- Connect the output of the first AND gate and the second AND gate to two separate inputs of an OR gate.
- The output of the OR gate will be the result of the expression.

3. Diagram:
- Here's a diagram of the circuit for the expression "NOT X AND Y OR NOT Y AND Z":
- (NOT X AND Y) OR (NOT Y AND Z)

![Expression 3 Circuit Diagram](https://i.imgur.com/utHhHXM.png)

4. Truth table:
- The truth table for "NOT X AND Y OR NOT Y AND Z" will show the output for each combination of inputs (X, Y, Z).

| X | Y | Z | NOT X AND Y OR NOT Y AND Z |
|---|---|---|-----------------------------|
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 1 |
| 0 | 1 | 0 | 0 |
| 0 | 1 | 1 | 1 |
| 1 | 0 | 0 | 0 |
| 1 | 0 | 1 | 1 |
| 1 | 1 | 0 | 1 |
| 1 | 1 | 1 | 0 |

I hope this explanation helps you understand how to design digital circuits based on Boolean expressions. Feel free to ask if you have any further questions!