What is the difference between

$\displaystyle \subset$ and $\displaystyle \subseteq$

Thank you very much

Printable View

- May 16th 2009, 04:58 AMAnimalxxvBasic Set theory...
What is the difference between

$\displaystyle \subset$ and $\displaystyle \subseteq$

Thank you very much - May 16th 2009, 05:38 AMjosipive
A,B sets

A $\displaystyle \subset$ B ---> this means that all elements from set A are also elements of set B where B is bigger set ( it has more elements (important) )

A $\displaystyle \subseteq$ B ---> this means that all elements from set A are also elements of set B where A can be equal to B

example: A = { 1, 2, 3 }

B = { 1, 2, 3 }

then this is true: A $\displaystyle \subseteq$ B , but this isnt A $\displaystyle \subset$ B

but if B is = { 1, 2, 3, 4 }

A $\displaystyle \subseteq$ B is true and this is also true A $\displaystyle \subset$ B - May 16th 2009, 05:41 AMPlato
You may not be pleased with this answer.

There is no precise difference; it generally depends upon the author and the textbook.

In general, $\displaystyle \subset $ stands for a proper subset and in that case this symbol $\displaystyle \varsubsetneq $ is also used.

That is, it is a subset but not the entire superset.

For example, all of the following are correct usage:

$\displaystyle \left\{ {1,2,3} \right\} \subset \left\{ {1,2,3,4,5} \right\},\;\left\{ {1,2,3} \right\} \varsubsetneq \left\{ {1,2,3,4,5} \right\}\left\{ {1,2,3} \right\} \subseteq \left\{ {1,2,3,4,5} \right\}$

To be clear as to a set being a proper subset it is best to use this $\displaystyle A \varsubsetneq B$.

That makes it clear that $\displaystyle A$ is not $\displaystyle B$. - May 16th 2009, 05:43 AMAnimalxxv
Thanks very much :]

I'll thank you both

:]