$A = \{1, 2, 4, a, b, c\}$.
$\{\} ∉ A$ (true).
My solution for this question is true. Since $\{\}$ is not an element of $A$. But at college I showed this question to my teacher and he said it is false because $\{\}$ is a subset of $A$ not an element. What's the correct solution for this?
PS: Here is the question from the book circled in red.
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Mohamed Magdy
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1What are $a,b,c?$ – gammatester Oct 14 '18 at 15:00
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@gammatester elements in set A. – Mohamed Magdy Oct 14 '18 at 15:09
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1@MohamedMagdy elements in set $A$ indeed, and one of them could equalize ${}$, right? – drhab Oct 14 '18 at 15:16
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@drhab I updated my question. – Mohamed Magdy Oct 14 '18 at 18:21
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I preassume that the symbols $1,2,4$ do not denote the empty set.
Then: $$\{\}\notin A\text{ is true if }a\neq\{\}\text{ and }b\neq\{\}\text{ and }c\neq\{\}$$
Otherwise it is false.
If nothing is known about $a,b,c$ then you should state that the statement is not true in general.
Further $\{\}$ is indeed a subset of $A$ but that is not relevant here.
drhab
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Why didn't you assume that a, b, c are English alphabet characters? – Mohamed Magdy Oct 14 '18 at 18:30
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Because in set-theory everything is a set, so $a,b,c$ must be notations for sets. Your update has no effect on my answer: it is not true in general. – drhab Oct 14 '18 at 18:41
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Well, then I disagree with the author. Let your teacher read this and ask him/her what he/she thinks of this. – drhab Oct 14 '18 at 19:12
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No, ${}\notin A$ is not true in general. That means that it is not true under all possible conditions. That depends on $a,b,c$. Also the statement is not false in general. You simply have not enough information and cannot say things as "it is true" or "it is false". – drhab Oct 14 '18 at 19:33
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The empty set is a subset of every set.
For the set $A$ In the question , the empty set is not an element of the set.
Mohammad Riazi-Kermani
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You are correct but in general we do not denote the empty set by $a$ – Mohammad Riazi-Kermani Oct 14 '18 at 18:51
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…he said it is false because $ \{ \} $ is a subset of $ A $ not an element.
If it's, as he said, not an element and "$ \notin $" means "is not an element of", then it's obviously true.
Chai T. Rex
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The fact that ${}$ is a subset of $A$ does not exclude that ${}$ is an element of $A$. – drhab Oct 14 '18 at 18:49
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The real question is: "is ${}$ not an element of $A$?" It is not: "is the instructor correct?" – drhab Oct 14 '18 at 18:54
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Yes, but there was confusion on the part of the student due to what the instructor said, so it was a question and clarifying that should help somewhat. – Chai T. Rex Oct 15 '18 at 17:38