Are the elements $c$ and $\{c\}$ the same in a set?

Question

I have a problem that asks me this:

If $A = \{a, b, c\}$ and $B = \{b, \{c\}\}$, is $B$ a subset of $A$?

What I'm confused about is if you treat $\{c\}$ and $c$ as the same elements.

I'm sort of confident that $\{c\}$ is distinct from $c$ and this would not be a subset, but I want to make sure.

Answer 1

I'm sort of confident that $\{c\}$ is distinct from $c$ and this would not be a subset, but I want to make sure.

Be more confident.   It is so in axiomatic set theory that a set cannot be a member of itself.

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As others have pointed out, it is possible that $a=\{c\}$, or even $b=\{c\}$; so we should make that cavat.$$\{b,\{c\}\}\nsubseteq \{a,b,c\}\text{ unless }a=\{c\}\text{ or }b=\{c\}$$

Answer 2

No it is not, since "$c$" in A is an element which different from $\{c\}$ in B where it is a set.

Answer 3

$\{c\}$ is a set containing the element $c$, one is a subset of $A$: $\{c\} \subset A$, and the other is an element in $A: c\in A.$

So no, $B = \{b, \{c\}\}$ is not a subset of $A = \{a, b, c\}.$ $B$ is a set which contains an element of $A$ and a subset of $A$. But since $\{c\} \notin A$, $B$ cannot be a subset of A.