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))

The exact constraints of the problem are as follows:

1. Each member of the Monetary Committee has a button to push: This means that each committee member (x, y, z) can push their respective buttons, independently.

2. The vice president or the president has a button to push (at least one of them—or both—have a button to push): This means that either the vice president (vp) or the president (p), or both, can push their buttons.

3. The safe opens only if a combination of the president, the vice president, and a number of the committee members push the button: This means that for the safe to open, a specific combination of inputs needs to be satisfied.

To design the safe circuit and create a truth table, we need to create a logical expression that represents the conditions mentioned above.

A possible approach to designing the safe circuit is to use the logical OR operator (represented by "+") to allow multiple inputs to be considered. We should also consider using the logical AND operator (represented by ".") to ensure that the specific combination of inputs is satisfied.

Based on the given constraints, one possible Boolean expression for the safe circuit is as follows:

f = (p + vp) . (x + y + z)

Explanation for the creation of the safe circuit:

The first part of the expression, (p + vp), represents the condition that either the president or the vice president (or both) need to push their buttons. This is achieved through the logical OR operator (+).

The second part of the expression, (x + y + z), represents the condition that a number of committee members need to push their buttons. Again, this is achieved through the logical OR operator (+).

Finally, the two parts of the expression are connected with the logical AND operator (.) to indicate that both conditions need to be satisfied for the safe to open.

Truth table:

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 | 0
0 | 1 | 1 | 1 | 0 | 0
0 | 1 | 1 | 1 | 1 | 0
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 | 0
1 | 0 | 1 | 1 | 0 | 0
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

In the truth table, "1" represents a true or pushed input, while "0" represents a false or not pushed input. The last column, "f," represents the output of the safe lock (1 = open, 0 = locked) for each input combination.