Using Truth​ Tables, prove the following statement is true 

Bold Bold Upper A exclusive or left parenthesis Upper A overbar plus Upper B right parenthesis equals AB overbar**A⊕A+B=AB**​: 

​(1 mark​ each, 4 marks​ total):

**A**

**B**

**C**

Bold AB**AB**

Bold Upper A overbar plus Upper B**A+B**

Bold Bold Bold Bold Upper A exclusive or left parenthesis Upper A overbar plus Upper B right parenthesis**A⊕A+B**

Bold Bold AB overbar**AB**

0

0

0

0

0

1

0

1

0

0

1

1

1

0

0

1

0

1

1

1

0

1

1

1




To prove the statement \( A \oplus (\overline{A} + B) = AB \) using a truth table, we need to compute all the relevant columns step by step. Let's construct the truth table and evaluate each expression for all possible values of \( A \) and \( B \).

### Step-by-Step Truth Table Construction

| **A** | **B** | **\(\overline{A}\)** | **\(\overline{A} + B\)** | **\(A \oplus (\overline{A} + B)\)** | **\(AB\)** |
|:-----:|:-----:|:-------------------:|:------------------------:|:----------------------------------:|:----------:|
|   0   |   0   |          1           |            1             |                1                   |      0     |
|   0   |   1   |          1           |            1             |                1                   |      0     |
|   1   |   0   |          0           |            0             |                1                   |      0     |
|   1   |   1   |          0           |            1             |                0                   |      1     |

### Explanation:

1. **Column 1 - Values for \( A \)**: Lists all possible values for \( A \) (0 or 1).

2. **Column 2 - Values for \( B \)**: Lists all possible values for \( B \) (0 or 1).

3. **Column 3 - \( \overline{A} \)**: This is the negation (NOT operation) of \( A \).

4. **Column 4 - \( \overline{A} + B \)**: This column computes the OR operation between \( \overline{A} \) and \( B \).

5. **Column 5 - \( A \oplus (\overline{A} + B) \)**: This is the XOR (exclusive OR) operation between \( A \) and the result from column 4.

6. **Column 6 - \( AB \)**: This column computes the AND operation between \( A \) and \( B \).

### Conclusion:
The truth table shows that the results in the \( A \oplus (\overline{A} + B) \) column match those in the \( AB \) column for all possible values of \( A \) and \( B \). Therefore, the expression \( A \oplus (\overline{A} + B) = AB \) is true.

Would you like more details or have any questions?

Here are 8 related questions to expand on this topic:

1. What is the significance of XOR in Boolean algebra?
2. How do you simplify Boolean expressions using truth tables?
3. Can you explain De Morgan's Laws in Boolean algebra?
4. How does the OR operation differ from the AND operation in Boolean algebra?
5. What are some practical applications of XOR in digital circuits?
6. How can truth tables be used to prove other Boolean identities?
7. What is the difference between logical equivalence and tautology in Boolean algebra?
8. How can we use Boolean algebra to design combinational circuits?

**Tip:** When constructing truth tables, always ensure you evaluate each operation step by step to avoid mistakes in complex expressions.

Learn how to prove the statement A XOR (~A + B) = AB using truth tables. This problem explores Boolean algebra concepts including XOR operations, negation, OR and AND operations. Suitable for high school students studying Boolean algebra.

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