# Set, Empty Set

• Nov 29th 2013, 07:46 AM
Hartlw
Set, Empty Set
Set: A named collection of objects.

Empty Set ( {}, or Ř ): Universal name (container) for a set with no objects. The empty set acquires a specific name when objects are added to it.

Example:
Let {} plus a,b,c be A={a,b,c}.
If a,b,c removed, A={}.

Named: identified, designated.
• Dec 2nd 2013, 11:19 AM
Hartlw
Re: Set, Empty Set
Some more thoughts on OP.

A set is defined by its contents, described or listed.

Named in broadest sense: denote, designate, identify, refer to, etc.

A common way of naming a set is to use {}. “{a,b,c}” is the name of {a,b,c}.

A is a subset of B if A only contains elements of B doesn’t work for {}, except in the negative sense, and doesn’t prove existence.

A is a subset of B if you can get A by removing none or more elements of B establishes existence of {}, gives that all {} are identical, and shows the empty set is a subset of every set.
• Dec 2nd 2013, 12:42 PM
Plato
Re: Set, Empty Set
Quote:

Originally Posted by Hartlw
Some more thoughts on OP.

A set is defined by its contents, described or listed.

Named in broadest sense: denote, designate, identify, refer to, etc.

A common way of naming a set is to use {}. “{a,b,c}” is the name of {a,b,c}.

A is a subset of B if A only contains elements of B doesn’t work for {}, except in the negative sense, and doesn’t prove existence.

A is a subset of B if you can get A by removing none or more elements of B establishes existence of {}, gives that all {} are identical, and shows the empty set is a subset of every set.

$A\subseteq B\text{ if and only if }(\forall x)[ x\in A\Rightarrow x\in B]$
• Dec 2nd 2013, 01:49 PM
Hartlw
Re: Set, Empty Set
Quote:

Originally Posted by Plato
$A\subseteq B\text{ if and only if }(\forall x)[ x\in A\Rightarrow x\in B]$

The empty set has no members. I know, the negative proof, because A is not false it is true. But you have to prove that exhausts the options, ie,
You haven't shown the empty set exists. Std Def: Set is a collection of objects. No objects, no set.

EDIT I should have been more specific. There is no problem with your definiton for a non-empty set. It's the empty set that is a problem.

EDIT Actually, there is a problem with your standard definition of subset. {a,b} is a subset of {a,b,c}. Are a,b a subset of {a,b,c}? Why not? They are a collection of objects so they satisfy standard definition of a set. a,b are simultaneously members of {a,b,c} and a subset of {a,b,c}. This thread addresses this.
• Dec 3rd 2013, 08:16 AM
Hartlw
Re: Set, Empty Set
I misconstrued “definition.”

A set is a collection of objects does not mean a collection of objects is a set. You have to specify the collection of objects as a set as in “consider the set a,b,c,” which is not the same as “consider the elements a,b,c.”

That leaves the empty set, which is not a collection of objects.
Defining a subset by removing objects from an existing set gives the existence of the empty set, and that the empty set is a subset of all sets. The standard definition of a subset given by Plato above then follows as a theorem.

In other words, A is a subset of B if you can get A by removing elements from B. Beautiful, except that the empty subset still does not satisfy the definition of a set as a collection of objects. Back to the first two posts.
• Dec 3rd 2013, 09:43 AM
Hartlw
Re: Set, Empty Set
Quote:

Originally Posted by Hartlw
I misconstrued “definition.”

A set is a collection of objects does not mean a collection of objects is a set. You have to specify the collection of objects as a set as in “consider the set a,b,c,” which is not the same as “consider the elements a,b,c.”

That leaves the empty set, which is not a collection of objects.
Defining a subset by removing objects from an existing set gives the existence of the empty set, and that the empty set is a subset of all sets. The standard definition of a subset given by Plato above then follows as a theorem.

In other words, A is a subset of B if you can get A by removing elements from B. Beautiful, except that the empty subset still does not satisfy the definition of a set as a collection of objects. Back to the first two posts.

"Beautiful, except that the empty subset still does not satisfy the definition of a set as a collection of objects." No problem, it is a definition and exists by construction. So strike last sentence: "back to the first two posts."