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 determine the conditions under which the safe will open.
1. Each member of the Monetary Committee has a button to push.
Constraint: x, y, z can take on values 0 or 1, representing whether the respective committee member has pushed their button or not.
2. The vice president or the president has a button to push (at least one of them—or both—have a button to push).
Constraint: vp and p can take on values 0 or 1, representing whether the vice president and the president have pushed their buttons or not. At least one of them should be 1.
3. The safe opens only if a combination of the president, the vice president, and a number of the committee members push the button.
Constraint: The safe will open if and only if the president (p) or vice president (vp) or both (p * vp) have pushed their buttons, AND at least one committee member (x + y + z) has pushed their button.
Based on these constraints, we can design the safe circuit using logical gates.
1. Create an OR gate with inputs p and vp. This gate ensures that either the president or the vice president or both have pushed their buttons.
2. Create an OR gate with inputs x, y, and z. This gate ensures that at least one committee member has pushed their button.
3. Connect the output of the OR gate from step 1 and the OR gate from step 2 to an AND gate. This gate combines the conditions of the president/vice president and the committee members.
4. The output of the AND gate is the safe lock signal (f). If this output is 1, it means the safe should be opened; otherwise, it remains locked.
To complete the truth table, we need to consider all possible combinations of inputs p, vp, x, y, and z and evaluate the output f using the Boolean expression we derived.
Here is the truth table:
| Input (p, vp, x, y, z) | Output (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 | 1 |
| 0 1 0 1 0 | 1 |
| 0 1 0 1 1 | 1 |
| 0 1 1 0 0 | 1 |
| 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 | 1 |
| 1 0 0 1 0 | 1 |
| 1 0 0 1 1 | 1 |
| 1 0 1 0 0 | 1 |
| 1 0 1 0 1 | 1 |
| 1 0 1 1 0 | 1 |
| 1 0 1 1 1 | 1 |
| 1 1 0 0 0 | 1 |
| 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 |
The Boolean expression for the safe lock (f) can be derived by analyzing the required conditions:
f = (p * vp) * (x + y + z)
This expression ensures that the safe remains locked (f = 0) unless at least one committee member's button is pressed (x, y, or z), and either the president's button (p) or the vice president's button (vp), or both are pressed.