# A null set is a subset of every set

1. May 26, 2014

### 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!

2. May 26, 2014

### 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?

3. May 26, 2014

### Staff: Mentor

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

from wikipedia article:

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

4. May 26, 2014

### chemistry1

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?

5. May 27, 2014

### jbriggs444

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.

6. May 27, 2014

### HallsofIvy

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.

7. May 27, 2014

### 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?

8. May 28, 2014

### johnqwertyful

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.