I want to know the difference in nuances of ⊂.

Question

Please tell me the difference between 1⊂A and {1}∈A. Please tell me the difference in nuances.

Answer 1

I'm not quite sure what you mean by nuances. Both these expressions have a very precise meaning. $x\in A$ means that the element $x$ is within $A$. For example, if $A = \{1, 2, 3, 4\}$ then $\{1\} \notin A$ but if $A = \{\{1\}, 2, 3, 4\}$ then $\{1\} \in A$.

On the other hand, $A\subset B$ simply means every element in $A$ is an element of $B$. It may very well be the case that $A$ is not an element of $B$. For instance, $\{1,3,5,6\} \subset \{1,2,3,4,5,6,7,8,9\}$, but there is no element $\{1,3,5,6\}$ in the second set.

In your example, when you mention $1\subset A$, it means you are treating $1$ as a set. If we take the Von Neumman construction of natural numbers, then $1 = \{\varnothing\}$ so if $1\subset A$, it means one of the elements of $A$ is the empty set.