# I If set A subset of B, and B of C, it does not necessarilly f

1. Nov 4, 2016

### Logical Dog

Continiung from the title, it does not necessarilly follow that A will be a subset of C. I knew this for a long time, but I am unable to understand why. Elias zakons notes gave an example'

here is the example:

This may be illustrated by the following examples.
Let a “nation” be defined as a certain set of individuals, and let the United
Nations (U.N.) be regarded as a certain set of nations. Then single persons are
elements of the nations, and the nations are members of U.N., but individuals
are not members of U.N. Similarly, the Big Ten consists of ten universities,
each university contains thousands of students, but no student is one of the
Big Ten. Families of sets will usually be denoted by script letters: M, N , P,
etc

could anyone give other examples, more mathematical object oriented?

2. Nov 4, 2016

### Krylov

I don't understand this. You mean to say that $A \subseteq B$ and $B \subseteq C$ does not imply $A \subseteq C$? I am confused.

3. Nov 4, 2016

### Logical Dog

yes. Have any more examples in terms of types of numbers?

4. Nov 4, 2016

### micromass

Staff Emeritus
You mean that if $A\in B$ and $B\in C$ that not necessarily $A\in C$?

5. Nov 4, 2016

### Logical Dog

Yes^^^ apologies for not being clear

6. Nov 4, 2016

### micromass

Staff Emeritus
No, since what krylov said is true. You seem to think it's not true for some weird reason.

7. Nov 4, 2016

### Logical Dog

No no. sorry, I meant if
1. a is a subset of B
2. and B is a subset of C,
3. it does not necessarily mean that A is also a subset of C

Would you be having any examples, in terms of numbers or just in general?

8. Nov 4, 2016

### micromass

Staff Emeritus
If 1 and 2 are true that it does necessarily mean that A is a subset of C.

9. Nov 4, 2016

### Logical Dog

Ok. Sorry for the confusion, I think I will go sleep and read it again.

10. Nov 4, 2016

### micromass

Staff Emeritus
You should revise the difference between subset and element.

11. Nov 4, 2016

### Logical Dog

No i got it, mixed up the symbols. sorry,

So if A is in B, and B is in C, A will most likely not be an element of C. As C = {{A}}

But subsets are always nested.

12. Nov 4, 2016

### micromass

Staff Emeritus
Here is a rather artificial example:
1. $2$ is an element of $P$, the set of prime numbers.
2. The set $P$ of prime numbers is an element of $\mathcal{P}(\mathbb{Z})$, the collection of all subsets of $\mathbb{Z}$.
3. But $2\in \mathcal{P}(\mathbb{Z})$ would mean that $2$ is a subset of $\mathbb{Z}$, which it is not.

13. Nov 4, 2016

### Staff: Mentor

Right.

A can be in C, however. C={{A},A}, done.

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