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I'm not sure how to think about the relationship between the statements

any input is LOW; 
no inputs are HIGH; and
the output of an AND gate is LOW.

The context is my textbook, the 11th edition of Digital Fundamentals by Thomas Floyd, which asks

The output of an AND gate is LOW when

(a) any input is LOW 
(b) all inputs are HIGH 
(c) no inputs are HIGH 
(d) Both (a) and (c)

and gives as the answer

(a)

I'm trying to think more logically and was hoping for some input.

First off, I understand r when s $\equiv $ if $s$, then $r$.

Let $$ q: \text{the output of an AND gate is LOW} \\ a: \text{any input is LOW} \\ c: \text{no inputs are HIGH} $$

Since the answer is $ (a) $, is it correct to express the idea as $ a \to q $?

Would it also be correct to observe that any time $c$ is true $a$ must also be true, and to express this relationship as $ c \to a $? By transitivity, doesn't that suggest $ c \to q$ as an answer also?

Yet $\neg(a \to c)$ because $a$ is a more restrictive condition than $c$. After all, the output of an AND gate can be LOW when $ a \land \neg c $ is true. So I'm not wanting to say that $ a \land c \to q$, even though $a \to q$ and $c \to q$.

I'm thinking that if the task is to select all propositions for which the if-then relationship is true, I want to select $a$ and $c$ but not $a \land c$ because $a \to q$ and $a \to q$ are true whereas $a \land c \to $q is not.

On the other hand, if the task is to select the proposition, $p$, that is always true whenever $q$ is true as well as sufficient for $q$, I'm thinking I want to select $a \to q$ because $q \to a$.

Since the answer is $(a)$, am I right to say that the spirit of the question is to identify which proposition, $p$, satisfies $p \iff $q? Only $p = a$ does.

WarrenTheRabbit
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  • Maybe the issue is only about the truth-table for AND, that has True only in case when all inputs are True. If High is True and Low is False, it is enough that one input has False (Low) and the gate will output False (Low). – Mauro ALLEGRANZA Mar 09 '22 at 12:45
  • In terms similar to your post: "A HIGH output results only if all the inputs to the AND gate are HIGH" and thus "A LOW output results when at least one input to the AND gate is LOW" – Mauro ALLEGRANZA Mar 09 '22 at 12:48
  • And we can rewrite is as follows: "The output of an AND gate is LOW iff any input is LOW". – Mauro ALLEGRANZA Mar 09 '22 at 12:58

1 Answers1

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In brief, you're asking, based on the available clues, what the author actually means by “when” in their sentence “The output of an AND gate is LOW when any input is LOW”.

Assuming no printing error, I agree with your analysis that the author means “if and only if (⟺)” there.

Equivalently, the author could have (accurately and unambiguously) written “precisely when” instead of “when”.

As written though, the MCQ's correct answer ought to be (d) instead of (a).

ryang
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