Find the Boolean expression E=E(x, y, z) corresponding to the truth tables: (a) T(E)=10001010; (b) T(E)=00010001
To find the Boolean expression E(x, y, z) corresponding to the truth table, we need to analyze the given binary sequences and identify patterns in the inputs (x, y, z) and corresponding outputs (E).
(a) T(E) = 10001010:
To convert this binary sequence into a truth table, let's assign the binary digits to their respective rows in the truth table:
| x | y | z | E |
|---|---|---|---|
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 1 | 1 | 0 |
| 1 | 0 | 0 | 1 |
| 1 | 0 | 1 | 0 |
| 1 | 1 | 0 | 1 |
| 1 | 1 | 1 | 0 |
Now, let's analyze the inputs and outputs to identify any patterns:
- The output E is 1 for the combinations (0, 0, 0), (1, 0, 0), and (1, 1, 0).
- The output E is 0 for the combinations (0, 0, 1), (0, 1, 0), (0, 1, 1), (1, 0, 1), and (1, 1, 1).
Based on these patterns, we can deduce the Boolean expression for E(x, y, z):
E = (x' * y' * z') + (x * y' * z') + (x * y * z')
(b) T(E) = 00010001:
Using a similar approach, let's convert the binary sequence into a truth table:
| x | y | z | E |
|---|---|---|---|
| 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 1 | 1 | 1 |
| 1 | 0 | 0 | 0 |
| 1 | 0 | 1 | 0 |
| 1 | 1 | 0 | 0 |
| 1 | 1 | 1 | 1 |
Analyzing the inputs and outputs, we observe the following patterns:
- The output E is 1 for the combinations (0, 1, 1) and (1, 1, 1).
- The output E is 0 for all other combinations.
Based on these patterns, we can deduce the Boolean expression for E(x, y, z):
E = (x' * y * z) + (x * y * z)+ (x * y * z') + (x * y' * z')