A null set is a subset of every set

[chemistry1]

Hi, I was wondering, how can a null set be a subset of other sets? Could anyone explain the idea in non technical terms, I'm just a beginner. :)

Thank you!

 

[HallsofIvy]

Do you agree that "if A is NOT a subset of B then there an element in A that is not in B"?

Do you see that, no matter what B is, the empty set cannot satisfy that?

 

[jedishrfu]

An interesting application of this is the construction of natural numbers via set theory:

Constructions based on set theory

A standard construction

A standard construction in set theory, a special case of the von Neumann ordinal construction,[7] is to define the natural numbers as follows:

Set 0 := { }, the empty set,
and define S(a) = a ∪ {a} for every set a. S(a) is the successor of a, and S is called the successor function.
By the axiom of infinity, the set of all natural numbers exists and is the intersection of all sets containing 0 which are closed under this successor function. This then satisfies the Peano axioms.
Each natural number is then equal to the set of all natural numbers less than it, so that
0 = { }
1 = {0} = {{ }}
2 = {0, 1} = {0, {0}} = {{ }, {{ }}}
3 = {0, 1, 2} = {0, {0}, {0, {0}}} ={{ }, {{ }}, {{ }, {{ }}}}
n = {0, 1, 2, ..., n−2, n−1} = {0, 1, 2, ..., n−2,} ∪ {n−1} = {n−1} ∪ (n−1) = S(n−1)
and so on.

from wikipedia article:

http://en.wikipedia.org/wiki/Natural_numbers

 

[chemistry1]

Do you agree that "if A is NOT a subset of B then there an element in A that is not in B"?

Do you see that, no matter what B is, the empty set cannot satisfy that?

Well, I do understand your first part. Now, for the second part, are you saying that because the empty set has no elements, it can't satisfy(be a subset) of B?

 

[jbriggs444]

Well, I do understand your first part. Now, for the second part, are you saying that because the empty set has no elements, it can't satisfy(be a subset) of B?

The part that the empty set cannot satisfy is the "then there [is] an element in A that is not in B", with the empty set playing the role of A.

 

[HallsofIvy]

Well, I do understand your first part. Now, for the second part, are you saying that because the empty set has no elements, it can't satisfy(be a subset) of B?

No, I'm saying the opposite of that. The first part was a condition for NOT being a subset of B. Since the empty set cannot satisfy it, the empty set must be a subset of B.

 

[Stephen Tashi]

chemistry1,

The first thing to understand is why a statement of the form "If A then B" is considered true when the statement A is false. (This is the way it is in mathematical logic - not in the way the man-in-the-street thinks about things.) There are many threads on the forum discussing this because most people find it a strange convention when they first encounter it. Do you understand why it is essential to have this convention in mathematics?

 

[johnqwertyful]

chemistry1,

The first thing to understand is why a statement of the form "If A then B" is considered true when the statement A is false. (This is the way it is in mathematical logic - not in the way the man-in-the-street thinks about things.) There are many threads on the forum discussing this because most people find it a strange convention when they first encounter it. Do you understand why it is essential to have this convention in mathematics?

I don't think it's all that strange.

"If I win the lottery, I'll give you a million dollars". But I didn't win the lottery, so I never lied to you.