What is the difference between
$\displaystyle \subset$ and $\displaystyle \subseteq$
Thank you very much
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
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 $\displaystyle B$.