1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Discrete Mathematics - Void Sets being Subsets of other Void Sets

  1. Aug 27, 2012 #1
    1. The problem statement, all variables and given/known data


    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.

    2. Relevant equations


    3. 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 {{∅}}.
  2. jcsd
  3. Aug 27, 2012 #2


    User Avatar
    Science Advisor

    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.
  4. Aug 27, 2012 #3
    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?
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook