Question

Design some logic with the following rules:
Two inputs are A and B
The output is C
The logic is as follows: If A and B are diferent values, the output is true
(1). In addiion, if A is true (1) then the output is true (1). Otherwise, the
output is false (0).
Build the Truth Table for this logic circuit. Using the Truth Table write the logic expression for this funcion.

Answer 1

Let's break down the logic rules you've provided into a clear truth table, and then from that table, we can derive the logical expression.

### Input Definitions:
- **A:** Input A (can be 0 or 1)
- **B:** Input B (can be 0 or 1)
- **C:** Output C (can be 0 or 1)

### Rules:
1. If A and B are different, C is true (1).
2. If A is true (1), then C is true (1). Otherwise, C is false (0).

### Truth Table:

Let's evaluate the cases based on the combinations of A and B.

| A | B | A ≠ B | A = 1 | Output C (based on rules) |
|---|---|-------|-------|--------------------------|
| 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 1 | 0 | 1 |
| 1 | 0 | 1 | 1 | 1 |
| 1 | 1 | 0 | 1 | 1 |

### Explanation of Output Column:
- When **A = 0** and **B = 0**, they are not different (A ≠ B is false), thus **C = 0**.
- When **A = 0** and **B = 1**, they are different, and since A is false, based on rule 2, **C = 1**.
- When **A = 1** and **B = 0**, they are different (A ≠ B is true), and since A is true, **C = 1**.
- When **A = 1** and **B = 1**, they are not different (A ≠ B is false) but since A is true (A = 1), **C = 1**.

### Final Truth Table:

| A | B | C |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 1 |

### Logic Expression:

Now let's express C logically. Based on the truth table, we derive that:

- C is 1 when:
- A is 1 (regardless of B) — this corresponds to the term \( A \)
- A and B are different, which can be represented as \( A \neq B \) or \( (A \land \lnot B) \lor (\lnot A \land B) \)

Combining these, we can simplify the expression into:
\[ C = A + (A \neq B) \]
Where \( (A \neq B) \) can be expressed using AND/OR gates as above.

Thus, the final expression can be represented in the standard Boolean algebra as:
\[ C = A + (\lnot A B + A \lnot B) \]
Or simplified as:
\[ C = A + (A \oplus B) \]
where \( \oplus \) denotes the XOR operation.

This captures the conditions specified and can be used to implement the logic circuit.