Question

the figure below shows a circuit composed of two logic gates. the output of the circuit is true.
image of logic circuit with or gate (inputs: false, a) and and gate (inputs: true, output of or gate), output: true
which of the following is a true statement about input a?
a) input a must be true
b) input a must be false
c) input a can be either true or false
d) there is no possible value of input a that will cause the circuit to have the output true

Explanation:

Step1: Analyze the AND gate

The AND gate has two inputs: one from the OR gate and the other is "True". The output of the AND gate is "True". Recall the truth table of an AND gate: \( \text{AND}(X, Y) \) is True only when both \( X \) and \( Y \) are True. Since one input to the AND gate is True, the other input (from the OR gate) must also be True (because \( \text{AND}(\text{True}, X) = \text{True} \) implies \( X = \text{True} \)).

Step2: Analyze the OR gate

The OR gate has inputs "False" and \( A \). The output of the OR gate is True (from Step 1). Recall the truth table of an OR gate: \( \text{OR}(X, Y) \) is True if at least one of \( X \) or \( Y \) is True. Here, \( X = \text{False} \), so \( \text{OR}(\text{False}, A) = \text{True} \) implies that \( A \) must be True (because \( \text{OR}(\text{False}, \text{False}) = \text{False} \) and \( \text{OR}(\text{False}, \text{True}) = \text{True} \)).

Answer:

A. Input A must be true