$A \subset B$ and $A \subseteq B$, what is the difference?

Question

So far I know that if $A \subset B$ then every element that is in $A$ will be also in $B$ but there will be at least one element in $B$ that is not in $A$

What about $A \subseteq B$? This site says that $A \subseteq B$ says that "$A$ is a subset of $B$. set $A$ is included in set $B$." But well, it's pretty much the same definition as $A \subset B$.

So my question is, $A \subset B$ and $A \subseteq B$, what is the difference?

Answer 1

ETA: Apparently, I'm fighting against the tide of convention. Use whatever definition the author of your book uses.

$A \subseteq B$ is $A \subset B$ or $A = B$. It's analogous to $a < b$ vs $a \leq b$.

Answer 2

It depends on convention. Some people use $A\subseteq B$ to mean $A$ is a subset of $B$ or $A=B.$ These people often use $A\subset B$ to denote proper subset. However, some use $A\subset B$ to mean that $A$ is a subset of $B$ or $A=B.$ These people typically use $A\subsetneq B$ to denote a proper subset.

In general, $A\subseteq B$ allows for the case of $A=B,$ and $A\subsetneq B$ means that $A$ is a strict subset of $B$, while $A\subset B$ is the primary source of ambiguity.

Answer 3

Annoyingly, these two symbols do mean exactly the same thing, unless the writer states otherwise. If you want to say that $A$ is a proper subset of $B$, you have to write $A\subsetneq B$.