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 $$\subseteq$$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 $$\subseteq$$. 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 $$\subseteq$$B as it is fine to say A $$\subset$$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 $$<$$ and $$\leq$$:

$$x\leq y$$ means "$$x<y$$ 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 $$\subseteq$$.