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$A = \{1, 2, 4, a, b, c\}$.
$\{\} ∉ A$ (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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    $\begingroup$ What are $a,b,c?$ $\endgroup$ Oct 14, 2018 at 15:00
  • $\begingroup$ @gammatester elements in set A. $\endgroup$ Oct 14, 2018 at 15:09
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    $\begingroup$ @MohamedMagdy elements in set $A$ indeed, and one of them could equalize $\{\}$, right? $\endgroup$
    – drhab
    Oct 14, 2018 at 15:16
  • $\begingroup$ @drhab I updated my question. $\endgroup$ Oct 14, 2018 at 18:21

3 Answers 3

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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.

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  • $\begingroup$ I updated my question. $\endgroup$ Oct 14, 2018 at 18:19
  • $\begingroup$ Why didn't you assume that a, b, c are English alphabet characters? $\endgroup$ Oct 14, 2018 at 18:30
  • $\begingroup$ 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. $\endgroup$
    – drhab
    Oct 14, 2018 at 18:41
  • $\begingroup$ The book author answered it true in the answers appendix. $\endgroup$ Oct 14, 2018 at 19:09
  • $\begingroup$ Well, then I disagree with the author. Let your teacher read this and ask him/her what he/she thinks of this. $\endgroup$
    – drhab
    Oct 14, 2018 at 19:12
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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.

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  • $\begingroup$ But who guarantees that e.g. $a\neq\varnothing$? $\endgroup$
    – drhab
    Oct 14, 2018 at 18:44
  • $\begingroup$ You are correct but in general we do not denote the empty set by $a$ $\endgroup$ Oct 14, 2018 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.

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  • $\begingroup$ The fact that $\{\}$ is a subset of $A$ does not exclude that $\{\}$ is an element of $A$. $\endgroup$
    – drhab
    Oct 14, 2018 at 18:49
  • $\begingroup$ That's true, but if the instructor is correct, then it's true. $\endgroup$ Oct 14, 2018 at 18:50
  • $\begingroup$ The real question is: "is $\{\}$ not an element of $A$?" It is not: "is the instructor correct?" $\endgroup$
    – drhab
    Oct 14, 2018 at 18:54
  • $\begingroup$ 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. $\endgroup$ Oct 15, 2018 at 17:38

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