Subsets of nested sets

Question

Excerpt from the textbook:

$M = \{a, \{b,c\}\}$. Which of the following statements are true?

1. $a \in M$
2. $b \in M$
3. $a \subseteq M$
4. $\{a\} \subseteq M$
5. $\{b,c\} \in M$
6. $\{b,c\} \subseteq M$

Solution: 1, 4 and 5 are correct

I do not get why 6 should not be a true statement or how it is supposed to differ from 4. If something is an element of a set is it not automatically a subset as well? Am I missing something or is this an error in the textbook?

Thank You

Answer 1

If something is an element of a set, it is not necessary a subset. For a sillier example if your set were the numbers $\{1,2,3\}$, then $1 \in \{1,2,3,\}$, but $\{1\} \subseteq \{1,2,3\}$.

For you example of the set $M$, the elements of $M$ are $a$ and $\{b,c\}$. That second element is itself a set, which I think is the source of your confusion. But yeah, $\{b,c\} \in M$, but $\{b,c\}\nsubseteq M$, and hopefully it'll help your understanding if I say that $\{\{b,c\}\} \subseteq M$ (set that contains the element $\{a,b\}$ is a subset of $M$).

Answer 2

$\{b,c\}$ is one element of $M$. Just as it is incorrect to say $a\subseteq M$, the given statement (6) is incorrect.

The following statements would be correct:

$\{b,c\} \in M$ (this is already statement 5); and

$\{\{b,c\}\}\subseteq M$

Answer 3

It is wrong to think that an element of a set is automatically a subset of that set (as you seem to do).

If $a\in A$ then you can draw the conclusion that $\{a\}\subseteq A$.

This because all elements of $\{a\}$ (there is only one which is $a$) are elements of $A$.

But we cannot conclude that $a\subseteq A$: there is no reason to think that all elements of $a$ are elements of $A$.

For instance take the set $A:=\{\mathbb N\}$ where $\mathbb N$ denotes the set of natural numbers. Here $A$ has exactly one element, which is $\mathbb N$. But $A$ does not contain any natural numbers, which would be the case if $\mathbb N$ would also be a subset of $A$.

Answer 4

A set $A$ is a subset of $B$ if for each element $x\in A$, it follows that $x\in B$. In question $6$ for example, $b\in \{b,c\}$ but $b\notin M$ (the only two elements of $M$ are $a$ and $\{b,c\}$). Thus $\{b,c\}\not\subseteq M$. In question $4$, $a\in \{a\}$ and it is also the case that $a\in M$. Hence $\{a\}\subseteq M$.