# Difference between membership and inclusion

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.

• The problem you are having is that you are not seeing the distinction between sets and the elements within a set. {2} is not the number 2. It is a set that has the number 2. h) sets the set with the number 2 is an element of a set of numbers. That is patently impossible. If {x: x is a number between 1 and 9} is a set of numbers then only numbers can be members. And {2} is not a number it is a set. – fleablood Mar 5 '18 at 20:31
• Thank you for your helpful response. However I'm still not entirely sure why ∅∈{a,b,c} is false though while ∅⊆{a,b,c} is true? As I don't know whether the empty set should be seen as a set or an element (or lack thereof) – smiles47 Mar 5 '18 at 20:45
• "I'm still not entirely sure why ∅∈{a,b,c} ". Well, what are the elements of {a,b,c}? They are a,b,c. Are any of a,b,c the same thing as the empty set? No, none of them are. So the emptyset is not an element of {a,b,c}. NOTHING should be seen as an element unless it actually is an element. Notice $\emptyset \in \{fred, george, \emptyset\}$ because the emptyset is in the set. And $\emptyset \not \in \{fred, george\}$ because the empty set is not in that set. It really is that simple. – fleablood Mar 5 '18 at 20:47
• I understand now why the empty set cannot be a member of {a,b,c}. hHowever, is that fact that the empty set can be included in the set {a,b,c} because every set includes the subset ∅? I.e., ∅⊆{a,b,c} (True) – smiles47 Mar 5 '18 at 20:57

## 3 Answers

$\{2\}$ 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 $9\}$

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.

• Thank you for your comment, however I'm not sure how this helps me to distinguish the difference between being a member and being included? – smiles47 Mar 5 '18 at 20:15
• The problem you hare having is not about being a member and being included (which are exactly the same thing) but about the difference between being an item, and being a set that contains the item. {2} is not 2. and 2 is not {2}. {2} is not a number. {2} is a set. 2 is not a set. 2 is a number. {2} is not a member of a set of numbers because {2} is not a number. – fleablood Mar 5 '18 at 20:31

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 $\{\{1, 2, 3, 4, 5\}, 1, 2, 3, 4, 5\}?)$

• Thank you for your reply! If I take my example then- {2}∈{x:x is a number between 1 and 9} (False)- would I be right in saying 2∈{x:x is a number between 1 and 9} is true (because 2, without the brackets around it, is now an element rather than a set?) – smiles47 Mar 5 '18 at 20:27
• It is false that $\{2\}$ is in the set because the set $\{2\}$ not the same as the number $2$. $\{x: \text{x is a number between 1 and 9}\}$ is a set of numbers $\{1, 2, 3, 4, 5, 6, 7, 8, 9\}$. However, you are right that $2 \in \{1, 2, 3, 4, 5, 6, 7, 8, 9\}$. (Now, if the set in question looked like $\{1, \{2\}, 3, 4, 5, 6, 7, 8, 9\}$, then we would have $\{2\}$ as a subset.) – Mauve Mar 5 '18 at 20:32
• Thank you for your helpful response. However I'm still not entirely sure why ∅∈{a,b,c} is false though while ∅⊆{a,b,c} is true? As I don't know whether the empty set should be seen as a set or an element (or lack thereof) – smiles47 Mar 5 '18 at 20:45
• The statement "Every element of $\{\}$ is an element of X" is vacuously true for all sets $X$, since there are no elements in $\{\}$. So the empty set is a subset of every set. (It is not an element of every set: $\{\} \in \{\{\}, 1, 2\},$ but the empty set is not an element of $\{1, 2, 3\}$) – Mauve Mar 5 '18 at 20:47