# Empty set as a subset?

1. Aug 28, 2012

### bonfire09

Ok im 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?

2. Aug 28, 2012

### SteveL27

3. Aug 28, 2012

### 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

4. Aug 28, 2012

### SteveL27

5. Aug 28, 2012

### bonfire09

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

6. Aug 28, 2012

### 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.

7. Aug 28, 2012

### bonfire09

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

8. Aug 28, 2012

### 1MileCrash

What???

9. Aug 28, 2012

### bonfire09

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

10. Aug 28, 2012

### 1MileCrash

With what you just wrote.

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

11. Aug 28, 2012

### 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

12. Aug 29, 2012

### 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.;

13. Aug 31, 2012

### 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?