Question

Use a truth table to prove the identity (A+B) (Ā + AB) = B

Answer 1

To prove the identity (A+B) (Ā + AB) = B using a truth table, follow these steps: create a truth table, fill in the truth values, calculate AB, calculate (A+B), calculate Ā+AB, compare (A+B) (Ā+AB) and B, and if they are the same for all rows, the identity is proven.

Step-by-step explanation:

To prove the identity (A+B) (Ā + AB) = B, we can use a truth table. The truth table is a table used in logic to determine the truth value of a compound statement for all possible combinations of truth values of its components. In this case, we have three variables: A, B, and Ā (the negation of A).

Here is the step-by-step process:

1. Create a truth table with columns for A, B, Ā, AB, (A+B), Ā+AB, and B.
2. Fill in the truth values for A, B, and Ā using 0s for false and 1s for true.
3. Calculate AB by performing the logical AND operation on A and B.
4. Calculate (A+B) by performing the logical OR operation on A and B.
5. Calculate Ā+AB by performing the logical OR operation on Ā and AB.
6. Compare the values of (A+B) (Ā+AB) and B in the last column.
7. If the values are the same for all rows, the identity is proven.

In this case, the truth table will show that (A+B) (Ā + AB) = B is true for all combinations of truth values of A and B, hence proving the identity.