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a) Ø ⊂ Ø False

b) Ø ⊂ {Ø} True

c)Ø ⊆ Ø True

d)Ø ⊆ {Ø} True

I am particularly confused with the difference of having {} and not having the braces because it seems that the braces make "b" true but without them "a" is false..Also what's the difference between "c" and "d"?

To be specific "a" is false because empty sets have no elements, but shouldn't "b" also be false since how can it be a proper subset if the only thing in the braces is the empty set, which makes it equal doesn't it?

Update: is b) true because {Ø} also contains an empty set making it {Ø, Ø}...?

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    $\begingroup$ a) is only false if you follow the convention that $\subset$ means $\subsetneqq$ (be warned, most [well, at least a lot] people consider $\subset$ to allow equality). $\{x\}$ is a set with one element, $x$. It so happens that $x$ could be the empty set. $\endgroup$ – Daniel Fischer Jul 1 '14 at 17:47
  • $\begingroup$ for b), its one element is the empty set, so are you saying that there's two empty sets in it? $\endgroup$ – user152573 Jul 1 '14 at 17:48
  • $\begingroup$ What does "in" mean? $S = \{\varnothing\}$ is a set with one element, which perchance is the empty set. Now $S$ is a nonempty set, hence the empty set is a proper subset of $S$, $\varnothing \subsetneq S$. For that, it is completely irrelevant what the element of $S$ is. Note that $\in$ and $\subset$ are completely different relations, even though in the vernacular "in" can be used when talking about either. But that's a bad idea, since it causes confusion. $\endgroup$ – Daniel Fischer Jul 1 '14 at 17:52
  • $\begingroup$ Are you saying that S becomes an empty set because its element is an empty set? I am confused with your explanation. $\endgroup$ – user152573 Jul 1 '14 at 18:00
  • $\begingroup$ The empty set (not $S$) is a proper subset of every nonempty set. For every element of the empty set (there is none) is also an element of the nonempty set (vacuous truth). But the nonempty set contains elements (any of its elements) that are not elements of the empty set (since the empty set contains no elements whatsoever). Thus the empty set is a proper subset of the nonempty set. $\varnothing \subsetneq \{ x\}$, regardless of what $x$ is. $S$ is a nonempty set. $\endgroup$ – Daniel Fischer Jul 1 '14 at 18:05
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If A is a subset of B this means all elements of A are elements of B. If A is a proper subset of B this means that all elements of A are elements of B but there is at least 1 elements of B which is not an elements of A.

In the notation you are using a subset is shown by ⊆ and a proper subset by ⊂.

If you check how these definitions apply to your questions

Every element of $\emptyset$ is an element of $ \emptyset$ because there are none - there is no element of $\emptyset $ which is not an element of $\emptyset$. This shows that (a) is false while (c) is true.

The empty set is in fact a subset of every set (not necessarily a proper subset) - there is no element in the empty set which is not in another set whatever the other set. So (b) is true. (d) is also true because the set {$\emptyset$} contains an element, $\emptyset$, whereas $\emptyset $ contains no elements.

Take note of previous comments on notation: it is more usual to notate a subset as $\subset$ and a proper subset as $\subsetneqq$.

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  • $\begingroup$ I didn't realize that there were different standards of the symbol since I am using a textbook by Grimaldi. $\endgroup$ – user152573 Jul 1 '14 at 18:20
  • $\begingroup$ What do you mean by "there is no element in the empty set which is not in another set ". $\endgroup$ – user152573 Jul 1 '14 at 18:40
  • $\begingroup$ These two statements are considered equivalent (1) "every element in A is in B" and (2) "there is no element in A which is not in B". If A is the empty set and B is any other set then statement (2) is true and therefore statement (1) is true and therefore the empty set is a subset of every set. $\endgroup$ – Tom Collinge Jul 1 '14 at 19:10
  • $\begingroup$ Regarding the notation, I personally prefer the ⊆, ⊂ version because of its immediate analogy with $\le$, $\lt$. But, the other is more wisely used so I stick with it (if there is one thing worse than a bad standard it is two "standards"). $\endgroup$ – Tom Collinge Jul 1 '14 at 19:39
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Hints:

  1. The empty set is a subset of any set, and is a proper subset of any set that is not empty.

  2. Any set that is a subset of the empty set is empty. Consequently the empty set does not contain proper subsets.

Notation $1$ connected with subset and proper subset: $\subseteq$ and $\subset$.

Notation $2$ connected with subset and proper subset: $\subset$ and $\varsubsetneqq$ (or $\subsetneq$).

Always make sure wich of the two is practicized.

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