Understanding Proper Subsets: An Explanation of the Empty Set in Set Theory

[bonfire09]

Ok I am a bit confused here. According to the definition of a proper subset means that everything in set A is in set b and a set always contains an extra pair of brackets. But in this example
C={∅,{∅}} why is this correct ∅ ⊆C instead of {∅} ⊆C for the first object?

 

[SteveL27]

I wrote up a detailed proof that the empty set is a subset of every set here

https://www.physicsforums.com/showthread.php?t=630957

in post #5. (I couldn't figure out how to link to my specific post, is that possible?).

 

[1MileCrash]

I'm a bit confused as to what you're asking. What do you mean extra brackets? A proper subset just means that if A is a proper subset of B, A is a subset of B and A =/= B.

If you're just asking why the empty set is always a subset, just look at what it would mean if it weren't. If the empty set weren't a subset of A, then that would mean the empty set contains some element that is not in A. But, that's impossible because the empty set has no elements.

Steve - right click the number and copy the link address, or just look at the format here:
https://www.physicsforums.com/showpost.php?p=4048220&postcount=5

 

[SteveL27]

Steve - right click the number and copy the link address, or just look at the format here:
https://www.physicsforums.com/showpost.php?p=4048220&postcount=5

Oh the numbers are the links! Thanks.

 

[bonfire09]

im asking is that if i say that ∅⊆C is this correct?

 

[1MileCrash]

Sure is.

Feel free to read that symbol as "Is a subset of OR is equal to." Since the empty set is a subset of anything, that statement is true and tautologous for any arbitrary C.

 

[bonfire09]

does this mean that what i said means that it points to the first empty set element within the set C?

 

[1MileCrash]

What?

 

[bonfire09]

how would i say that the first empty set in set C is a subset of C?

 

[1MileCrash]

With what you just wrote.

∅ is the empty set.
{∅} is.. "the set of the empty set."

 

[bonfire09]

oh ok. I was just a bit confused since you usually you put brackets around something when your saying that a set is a subset of another set such as this-

C={4,5,6) D={1,2,3,4,5,6}

{4,5,6}⊆{1,2,3,4,5,6}

But with the empty set I assume don't need to put brackets around it unless its a set within another set

 

[HallsofIvy]

Specifically, {∅} is NOT empty- it contains one element, the empty set. The set you give, C= {∅,{∅}} contains two elements, the empty set and the set whose only member is the empty set. Here it is perfectly correct to say that ∅$\subset$ C (the empty set is a subset of any set, as you say), {∅}$\subset$ C because ∅ is a member of C, and {{∅}}$\subset$ C because {∅} is a member of C.;

 

[bonfire09]

oh ok i get it but how about this one A={4,{5},6} and B={{5},6,7}. would A be a proper subset of B?