- #1

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## 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 {{∅}}.