Logic functions and circuit design

Explain what each row of a truth table represents and what it means for the output to be 1 in only some rows, using this example to support your explanation: an indicator light turns on when there is smoke (H = 1) and the alarm is activated (A = 1).

A car beeps if the door is open (P = 1) and the ignition is off (C = 1). Identify the inputs and output of the system and write its logic function using letters.

Three smoke sensors (A, B, and C) monitor an industrial unit, and the alarm must trigger when at least two of them detect smoke at the same time. Write the complete truth table for the system.

What is a logic function? Read these two and explain their meaning in words: $S = A \cdot B$ and $S = A + B$. Which gate implements each one?

A car alarm must sound when a door is open (P = 1) and the key is removed from the ignition (C = 1). Write the complete truth table and, using the rows with output 1, write the logic function.

A warning signal is required to sound when the tank is low (N = 1) or when the pump is stopped (B = 1) with the timer running (M = 1). Write the truth table with the three inputs and the logic function.

The function $S = A \cdot B + A \cdot \overline{B}$ is activated if 'A and B' or if 'A and not B'. Explain why it is simply equivalent to $S = A$ and what is saved in the circuit by noticing this.

Verify, by writing both truth tables, that $S = A \cdot (A + B)$ is equivalent to $S = A$. Explain the practical consequence of this result when building the circuit.

A student designs a home alarm system combining two sensors: door magnetic contact ($A = 1$ if open, $A = 0$ if closed) and volumetric motion radar ($B = 1$ if presence is detected). To ensure all situations are covered, he writes in his technical report:

"The alarm ($S = 1$) must trigger if there is motion with the door open ($A \cdot B$), and also if there is motion with the door closed ($\overline{A} \cdot B$). Therefore, the function is $S = A \cdot B + \overline{A} \cdot B$, which I will implement with two AND gates, one NOT, and one OR."

1. a) Prove by algebraic simplification and using the corresponding truth table what the actual minimum logic function of the system is.
2. b) Explain the physical meaning of the mathematical result obtained: does the door sensor $A$ provide any useful information to activate the siren? Which components from the initial setup are redundant?
3. c) Predict what would happen to the student's system if the door sensor $A$ breaks down and remains permanently disconnected (erroneous logic level): would the alarm stop sounding in the presence of an intruder or would security be compromised compared to the simplified circuit? Justify your answer.

A circuit has a NOT gate on input A and then an AND gate that receives that signal and input B. Write the logic function of the circuit and explain what situation it detects (which combination of inputs activates the output).

The security system of an industrial warehouse has three digital sensors: door opening ($P = 1$), window opening ($V = 1$), and system arming switch ($A = 1$ when surveillance is activated). The alarm ($S = 1$) must sound if the door or the window is opened, but only when the system is armed.

An installer connects the sensors according to the expression $S = P + V \cdot A$, arguing that "since the door is the main entrance, it must have a direct connection to the alarm annunciator".

1. a) Identify with which combination of inputs the technician's installation will fail and explain the serious security or false alarm problem it will cause in the warehouse.
2. b) Draw up the truth table that accurately models the requested specifications and obtain the canonical function as a sum of minterms.
3. c) Simplify the resulting function using the properties of Boolean algebra and describe the optimal electronic circuit, indicating which specific logic gates and how many inputs each one requires.

You want to design the smart lighting system for a communal hallway using three signals: motion detector ($M = 1$), twilight night sensor ($N = 1$), and manual maintenance switch ($I = 1$). The light ($L = 1$) must turn on automatically when it is night and motion is detected, or at any time if the manual switch is actuated.

1. a) Draw the truth table of the system and derive the simplified logic function $L(M, N, I)$.
2. b) The residents' association requests the addition of a "vacation mode" via a master switch $V$ ($V = 1$ vacation mode active). In this mode, automatic night-time activation must be completely disabled to save energy in case of drafts, but the manual switch $I$ must still be able to turn on the light for security personnel needs. Derive the new function $L_{vac}(M, N, I, V)$, justifying the modification without the need to construct a 16-row truth table.
3. c) In the initial circuit (without vacation mode), residents report that the light remains continuously on 24 hours a day without anyone pressing switch $I$ or anyone being in the hallway. Reasonably deduce which two independent physical faults in the circuit inputs could be causing this failure.