Understanding the Difference between \subseteq and \subset in Sets

[ayusuf]

If A \subseteq B does that mean A = B which means B = A because if A is a proper \subset of B then A does not equal B right. I am wrong right?

 

[VeeEight]

A <br /> \subseteq<br />B means that A is a subset of B. A could possibly equal B, but not in general.
For example, if A = {1} and B = {1, 2, 3} then A is clearly a subset of B as all elements of A are also elements of B. Here, A is a proper subset of B.

Usually, the symbol for a proper subset has a 'slash' through the horizontal line in the symbol <br /> \subseteq<br />. I can't seem to find it, however.

It is useful to note that some people use the symbols \subseteq and \subset to mean the same thing.

 

[ayusuf]

Yes exactly so if A \subseteq B then every element in A must be in B and if A does not equal B then A is a proper \subset of B.

 

[VeeEight]

Yes exactly so if A \subseteq B then every element in A must be in B

Yup, that is exactly what that means.

and if A does not equal B then A is a proper \subset of B.

Yes, if A\subseteq B and A does not equal B, then A is a proper subset of B.

 

[ayusuf]

But everytime A \subseteq B that must mean A = B right? If not please give me an example. Thanks.

 

[sylas]

But everytime A \subseteq B that must mean A = B right? If not please give me an example. Thanks.

The example was given to you in [post=2551076]msg #2[/post]. A = {1}, B = {1,2,3}.

 

[ayusuf]

Right so from that example it would be wrong to say that A \subseteq B but rather we should say A is a proper \subset of B because A \neq B.

 

[sylas]

Right so from that example it would be wrong to say that A \subseteq B but rather we should say A is a proper \subset of B because A \neq B.

No. It is completely correct to say A \subseteq B.

In the same way, it is completely correct to say 3 \leq 5.

 

[VeeEight]

Yes, in the example from message 2, A is a proper subset of B.
However, it is fine to say A <br /> \subseteq<br />B as it is fine to say A <br /> \subset<br />B.

The use of these two symbols are a matter of preference. Some professors will prefer to use one over the other but they both mean the same thing.

In the link http://en.wikipedia.org/wiki/Naive_set_theory#Subsets, the notation for proper subsets is in the last line of the second paragraph.

 

[ayusuf]

Okay I kind of get it. Thanks!

 

[vela]

Right so from that example it would be wrong to say that A \subseteq B but rather we should say A is a proper \subset of B because A \neq B.

A \subseteq B means "A is a subset of B"; A \subset B means "A is a subset of B and A is not equal to B." If A=B, it would be accurate to say A \subseteq B but not A \subset B. If A\ne B and A is a subset of B, either would be fine.

 

[Landau]

As sylas mentioned, this is analoguous to &lt; and \leq:

x\leq y means "x&lt;y or x=y".

A\subseteq B means "A\subset B or A=B".

(To deepen the analogy, they both define a partial order.)

Of course, with this explanation you have to know that it is implicit in A\subset B that A does not equal B.

 

[Johnson04]

I think you just need to check their respective definitions. A \subseteq B just means that \forall x\in A, x\in B. This definition does not say anything about the elements in B. In other words, \forall x\in B, it could be either in A or not in A. If \forall x\in B, implies x\in A, then A=B; if not, then A\not=B.

The definition of \subset is that \forall x\in A, x\in B, and \exists y\in B, such that y\not\in A. From this definition, we can see that actually, \subset is a special case of \subseteq.