I saw an answer for this question but I don’t understand yet, but what I concluded from it, is hat if $A \subset B$ then every element from $A$ is an element from $B$ but there are some elements in $B$ that are not in $A$, which means $B$ is bigger than $A$. Is it true?
And if $A \subseteq B$ so every element from both sets belong to the other one, which means always $A=B$ and it’s the same for $B \subseteq A$ ($A \subseteq B = B \subseteq A$).
Am I right? And please if there are any addition I would like to hear it.
Thanks!
$A⊂B$ is used by many authors to imply $A$ is a proper (or strict) subset of $B$, meaning that every element of $A$ is also an element of $B$, but that $A$ cannot be the same as $B$, i.e. there must exist at least one element in $B$ which is not in $A$.
$A⊆B$ implies $A$ is a subset of $B$, but may or may not be a proper subset. Hence every element of $A$ is in $B$, but there may or may not be elements in $B$ that are not in $A$.
To show two sets have exactly the same elements will be represented as $A=B$, which implies $A⊆B$ and implies $A⊂B$ is false. However, $A⊆B$ does not imply $A=B$, as if $A$ is a proper subset of $B$ then $A⊆B$ holds but $A=B$ does not.