Discrete Mathematics - Void Sets being Subsets of other Void Sets

[johnstobbart]

Homework Statement

Hello.

Here is the question:
Determine whether or not R is some sort of order relation on the given set X.

X = {∅, {∅}, {{∅}} } and R ε ⊆.

I can't seem to figure out why the ordered pairs given are what they are.

Homework Equations

None.

The Attempt at a Solution

What I first wrote out was:
R = { (∅, {∅}), ({∅}, {{∅}}), (∅, {{∅}}) }

Which is missing some ordered pairs. Also, my book says ({∅}, {{∅}}) is not an element of ⊆.

I tried to use my limited logic to understand the answer given, and this is what I got:

(∅, ∅) is an ordered pair because all the elements of ∅ are in ∅.
(∅, {∅}) is an ordered pair because ∅ is a member of {∅}.
(∅, {{∅}}) This causes some confusion. My book says the only member of {{∅}} is {∅}, but the first coordinate has to be a subset of the second coordinate. If ∅ is not an element of {{∅}}, how can it be a subset?
({∅}, {∅}) is an ordered pair because they are equal and subsets of each other.
({{∅}}, {{∅}}) same as above.

I don't understand why {∅} is not a subset of {{∅}}. {∅} is an element of {{∅}}, and should be a subset of it to my understanding because all the elements of {∅} are also within {{∅}}.

 

[HallsofIvy]

I think you are not distingishing between being a subset and being a member of a set.
{∅} is not a subset of {{∅}} because {∅} contains the empty set as a member and {{∅}} does NOT. "All the elements of {∅} are also within {{∅}}" is not true. The only member of {∅} is the empty set and that is NOT a member of {{∅}} because it only member is {∅}, not the empty set. The empty set is a subset of every set but not necessarily a member.

 

[johnstobbart]

Thanks for the reply HallsofIvy. That explains why ∅ is a subset of {{∅}}, while {∅} is not. ∅ is a subset of every set, while {∅} is not because that it is the set that contains only ∅. Is that correct?