# Is $\{\} ∉ A$ true or false, if $A = \{1, 2, 4, a, b, c\}$?

$$A = \{1, 2, 4, a, b, c\}$$.
$$\{\} ∉ A$$ (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.

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

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.

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

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.

• But who guarantees that e.g. $a\neq\varnothing$? Oct 14, 2018 at 18:44
• You are correct but in general we do not denote the empty set by $a$ Oct 14, 2018 at 18:51

…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.

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