In the country of Genovia, the president wants to ensure that the Monetary Committee can activate a device that opens the country’s safe. The safe system is to be activated by a device that obeys the following rules:
Each member of the Monetary Committee has a button to push.
The vice president or the president has a button to push (at least one of them—or both—have a button to push).
The safe opens only if a combination of the president, the vice president, and a number of the committee members push the button.
Complete the following:
Set the exact constrains of the problem.
Design the safe circuit.
Complete the corresponding truth table.
Explain your rationale on the creation of safe circuit.
Write the corresponding Boolean expression.
Specify the input and output variables and the two states of each.
Input:
p = president’s button (1 = pushed, 0 = not pushed)
vp = vice president’s button ( 1= pushed, 0 not pushed)
x, y, z = Monetary committees’ buttons (1 = pushed, 0 = not pushed)
Output:
f = Safe lock (1 = open, 0 = locked))
To set the exact constraints of the problem, we need to consider the given rules:
1. Each member of the Monetary Committee has a button to push.
2. The vice president or the president has a button to push (at least one of them—or both—have a button to push).
3. The safe opens only if a combination of the president, the vice president, and a number of the committee members push the button.
Based on these constraints, the safe circuit should be designed as follows:
1. Create a circuit with input variables: p (president's button), vp (vice president's button), x, y, z (Monetary Committees' buttons).
2. Use logical OR gates to ensure that either the president or the vice president (or both) have to push their buttons. This can be achieved by connecting the p and vp inputs to an OR gate.
3. Connect the output of the OR gate to AND gates along with the Monetary Committees' buttons (x, y, z). This ensures that a combination of the president/vice president and committee members is required to open the safe.
4. Connect the outputs of the AND gates to another OR gate to obtain the final output of the safe lock.
The corresponding truth table for the safe circuit is as follows:
| p | vp | x | y | z | f |
|-----|----|---|---|---|---|
| 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 1 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 1 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 1 | 1 | 1 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 1 | 1 | 0 |
| 0 | 1 | 1 | 0 | 0 | 0 |
| 0 | 1 | 1 | 0 | 1 | 1 |
| 0 | 1 | 1 | 1 | 0 | 1 |
| 0 | 1 | 1 | 1 | 1 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 1 | 1 | 0 |
| 1 | 0 | 1 | 0 | 0 | 0 |
| 1 | 0 | 1 | 0 | 1 | 1 |
| 1 | 0 | 1 | 1 | 0 | 1 |
| 1 | 0 | 1 | 1 | 1 | 1 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 1 | 1 |
| 1 | 1 | 0 | 1 | 0 | 1 |
| 1 | 1 | 0 | 1 | 1 | 1 |
| 1 | 1 | 1 | 0 | 0 | 1 |
| 1 | 1 | 1 | 0 | 1 | 1 |
| 1 | 1 | 1 | 1 | 0 | 1 |
| 1 | 1 | 1 | 1 | 1 | 1 |
Here's the rationale behind the creation of the safe circuit:
1. The OR gate with inputs p and vp ensures that either the president or the vice president (or both) have pushed their buttons, satisfying rule 2.
2. The AND gates connected to the Monetary Committees' buttons (x, y, z) ensure that a combination of the president/vice president and committee members is required, satisfying rule 3.
3. The final OR gate combines the outputs of the AND gates to determine whether the safe lock should be opened or not.
The corresponding Boolean expression for the safe circuit is:
f = (p + vp) * (x * y * z)
where '+' represents logical OR and '*' represents logical AND.