Difference between membership and inclusion

Question

I've taken the definition of membership to be the following:

Membership $A \in B: A$ is one of the members of $B$.

However, I'm not sure where to make the distinction between membership and inclusion, and hence I can't wrap my head around the solutions to the following questions:

"Say whether the following are true or false"

h. $\{2\}\in\{x:x$ is a number between $1$ and $9\}$ (False)

i. $\{2\}\subseteq\{x:x$ is a number between $1$ and $9\}$ (True)

and similarly,

n. $\emptyset\subseteq\{a,b,c\}$ (True)

o. $\emptyset\in\{a,b,c\}$ (False)

I am not sure why (h) and (o) are false but (i) and (n) are true, i.e. I don't see how the same element can be a subset but not a member of the same set.

Is it possibly because membership is only valid between an element and a set rather than a set and a set, while inclusion is valid between a set and a set?

I would appreciate any help in clarifying this, thank you.

Answer 1

$\{2\}$ and $2$ are entirely different things.

$\{2\}$ is a set that has $2$ as its only element-- it is a set and it is not a number. And $2$ is a number-- it isn't a set.

What are the members of $\{x: x$ is number between $1$ and $9\}$? Well those members are: $1,2,3,4,5,6,7,8,9$. Are any of them the same thing as $\{2\}$? Nope. Not a single one of those numbers between $1$ and $9$ is the set with $2$ as its only element. So $\{2\}\not \in \{x: x$ is number between $1$ and $9\}$.

Are any of those members the same thing as $2$; the number $2$? Yes, $2$ is the same thing as $2$. So $2 \in \{x: x$ is number between $1$ and $9\}$

Is $\{2\}$ a subset of $\{x: x$ is number between $1$ and $9\}$? Well, is $\{2\}$ a set? Yes. What are its members? Its member is $2$. What are the members of $\{x: x$ is number between $1$ and $9\}$? They are $1,2,3,4,5,6,7,8,9$. Are all of $2$ in the list $1,2,3,4,5,6,7,8,9$? Yes, it is.

So $\{2\}\subset \{x: x$ is number between $1$ and $9\}$.

Is $2$ a subset of $\{x: x$ is number between $1$ and $9\}$? Well, is $2$ a set? No, it is not. What are its members? It's not a set; it doesn't have any members.

So $2 \not \subset \{x: x$ is number between $1$ and $9\}$.

.....

"how the same element can be a subset"

An element can not be a subset at all.

$\{2\}$ is not an element of $\{x: x$ is number between $1$ and $9\}$

Answer 2

Take a look at these examples

$$2\in \{1,3,2\} $$

$$\{2\}\in \{\{1\},\{3,4\},\{2\}\} $$

$$\emptyset\in \{\{3,5\},\emptyset\} $$

a set belongs to a set of sets.

a set is included in a set which contains its elements.

Answer 3

Inclusion: If all objects inside set A are also inside set B, $A \subset B$.

Membership: If set/object A is itself inside set B, $A \in B$.

For example, let $A= \{1, 2, 3, 4, 5\}$.

$A \subset \{1, 2, 3, 4, 5, 6\},$

while $A \in \{\{1, 2, 3, 4, 5\}, 1, 2, 3\}$ (this could also be written $A \in \{A, 1, 2, 3\}).$

(Now... what can we say about $A$ and the set $\{\{1, 2, 3, 4, 5\}, 1, 2, 3, 4, 5\}?)$