Set theory: difference between belong/contained and includes/subset?

Question

This is a total noob question.

I am reading Naive Set Theory by Paul R. Halmos, and I'm having difficulty to understand something which seems to be trivial.

In the first chapter he writes:

If $x$ belongs to $A$ ($x$ is an element of $A$, $x$ is contained in $A$), we shall write

$x\in A$

I understand this.

Then, he write:

If $A$ and $B$ are sets and if every element of $A$ is an element of $B$, we say that $A$ is a subset of $B$, or $B$ includes $A$, and we write:

$A \subset B$

I understand this too.

Then he says:

The working of the definition implies that each set must be considered to be included in itself ($A \subset A$); this fact is described by saying that set inclusion is reflexive.

I understand this too.

But then:

Observe that belonging ($\in$) and inclusion ($\subset$) are conceptually very different things indeed. One important difference has already manifested itself above: inclusion is always reflexive, whereas it is not at all clear that belonging is ever reflexive. That is: $A \subset A$ is always true; is $A\in A$ ever true? It is certainly not true of any reasonable set that anyone has ever seen.

And this is where I don't think I understand anything. There is not more elaboration on this point in the text.

I tried to skip this but it seems it is quite fundamental for understanding what follows in the book.

Could someone explain what is meant here?

Answer 1

Whenever you come across something like this and it trips you up, you might want to look at particular examples. For instance, consider $\{4\}$. $\{4\} \subset \{4\}$, but $\{4\} \in \{4\}$ is false, since the only member of $\{4\}$ is $4$, not $\{4\}$. It may have tripped you up that "includes" and "contains" in everyday language usually qualify as synonyms. They don't here, and the terms get defined by the definitions for $\in$ and $\subset$. You might want to prove that $A \subset A$ for any set $A$ as it can get proven in a line or two.

Answer 2

The point is that every set is a subset of itself, namely $A\subseteq A$ - always.

However $\in$ does not have this property, for example $\varnothing$ has no elements, in particular $\varnothing\notin\varnothing$. However $\varnothing\subseteq\varnothing$.

To make matters worse, $\varnothing\in\{\varnothing\}$ as well $\varnothing\subseteq\{\varnothing\}$. However $\varnothing\neq\{\varnothing\}$ since the empty set has no elements and the singleton $\{\varnothing\}$ has an element.

In the axiomatic approach to set theory, the commonly used axioms of ZF dictate that $A\notin A$ for every set $A$. This is a result of something called the axiom of regularity, or axiom of foundation. However there are useful instances of non-well founded set theory in which some sets have the form $x=\{x\}$. For more information on that: When is $x=\{ x\}$?

Regardless to that, it is always the case that $A\subseteq A$, and always the case that for some $B$ we have $B\notin B$.

Answer 3

If something belongs to set then it means thats it is an element of that set as a whole but if a set is a subset of another set then it means all the elements of that set belong to the set to which that set is a subset.

Ex: Lets take two sets $A=\{1,2,3\}$ & $B=\{x\mid x \text{ is a natural number and } x<5\}$
Here, clearly evey element of set $A$ is an element of set $B$ hence we can say $A$ is a subset of $B$ but we can't say $A$ belongs to $B$ as set $A$ as a whole is not an element of set $B$.

Ex 2: $A=\{1,2,3\}$ & $B=\{\{1,2,3\},4,5\}$
Here set $A$ is an element of set $B$ itself. Hence we can say that $A$ belongs to $B$ but here $A$ is not a subset of $B$ as any individual element of $A$ won't be an element of set $B$.

Answer 4

• This from "ASK DR. MATH". All the CREDITS go to him! I found this very helpful so please take your time and efforts to look at his: Link

I will copy and paste what he said because it was extremely helpful in explaining the difference:

A SET is like a "bunch" or "collection" or "group" of things. An example is the set of girls in your daughter's school class. Another example is the set of all 2-digit perfect square numbers greater than your age. That is written { 49, 64, 81 }. This is a finite set so you can list the things in it. The 3 things in it are its ELEMENTs or its MEMBERs.

The order in which you list the elements makes no difference. For instance, {2, 3, 5, 7} is considered to be exactly the same set as {5, 7, 3, 2}. Whether you list them in numerical order or alphabetical order, this is still the set of all one-digit prime numbers.

Some sets are infinite, like the set of all even numbers greater than your age, which can be written

{ 2*N | N is a whole number and N > 23 }.

You say this "The set of all numbers of the form 2*N where N is a whole number and N is greater than 23". You can also write this set as { 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, ..... } .

The INTERSECTION of 2 sets is another set. The members of the INTERSECTION set have to be in both of those 2 sets. Think of the sets of numbers I mentioned in the previous paragraph. The only number that is in both sets is 64, so the intersection is { 64 } which is a set with exactly one element.

What about the UNION? Whereas the INTERSECTION of 2 sets contains the elements that are in BOTH of the 2 sets, the UNION of 2 sets contains the elements in EITHER one of the 2 sets. Here is an example: If the first set is { 2, 3, 4, 5, 6 } and the second set is { 4, 5, 6, 7 } then the intersection of the 2 is { 4, 5, 6 } and the union of the 2 is { 2, 3, 4, 5, 6, 7 }.

Let's see, what's left? SUBSET. It's sort of what it sounds like.
Let's do this with another example. I will specify 2 sets, called set A and set B, as follows:

A is the set of all girls in your daughter's class at school.

B is the set of all girls in your daughter's class at school whose first name begins with a vowel.

It is clear that any member of B is also a member of A, just by the way these 2 sets are defined. This is what we mean by saying that B is a SUBSET of the set A. I don't know your daughter's name, so I'm not going to be very accurate here, but let's see how this could turn out. I'll present two possibilities for the set A. I'll assume your daughter is Francesca.

A = { Francesca , Maria , Anita , Jean , Irene }
A = { Francesca , Maria , Donna , Jean , Kendra , Hillary }

If the top version is the true one, then B is { Anita , Irene }. If the bottom version is the true one, then B is ..... wait a minute here! .... there are NO NAMES that begin with a vowel. Precisely, so B is still a perfectly good set which just happens not to have ANY members. This is called the EMPTY SET. The empty set is a subset of all sets. Strange but true.

This is a start for you. At your Public Library in the math section they usually have books at many levels of learning. Often the reference librarian can steer you to something useful if you describe what you are looking for. Good luck and have fun.

-Doctor Mike, The Math Forum Check out our web site! Link

Answer 5

The best way to understand the differences is by describing the relation's properties.

• Belongs to is not reflexive, an element can not belongs to itself; it is not simetric, an element belongs to a set but a set doesn't belong to an element; it is not transitive, an element belongs to a set and this set belongs to another set the element don't belong to the last set.

• Contains is reflexive, a set contains itself; it is not symmetric, if a set contains another set, this set does not necessarily contain the first; it is transitive if a set contains another set that has another set, the first set contains the last set.

• Equal is reflexive, a set is equal to itself; it is symmetric, if a set is equaled to another set, the anther set is equal to the set; it is transitive if a set is equal to another set and this set is equal to another set, all sets are equals.

Answer 6

The problem is artificially created by the author of the book. There are two different concepts, an element and a set, which should not be confused.

An element is like an indivisible particle. Example: A letter is an element of a big set called alphabet. For real numbers, an element is a particular value, for example, $0.1$.

A set is like a bag with multiple (but unique!) elements inside or an empty bag with no elements at all. In our alphabet example, a set is any collection of unique letters. It means that we choose some letters out of 26 letters and put them in our bag (there are $2^{26}$ ways of doing so and therefore $2^{26}$ different sets). For real numbers, a line segment $[0,1]$ is an example of a set. It is a "bag" with all values in this interval. Of course, the number of possible choices in this case is infinite.

The concept belongs compares an element and a set. The corresponding symbol "$\in$" must have an element on the left hand side and a set on the right hand side: $x \in A$ means that an element $x$ is present in the bag $A$. For example, $a\in\{a,e,i,o,u\}$ (our alphabet example) and $0.1 \in [0,1]$ (our example with real numbers).

The concept includes compares two sets. Therefore it should have sets on both sides of the symbol "$\subset$". It says that if some element is present in a set on the left hand side, it is also present in the set on the right hand side. For example, $[0,0.5]\subset[0,1]$ means that if a point (element) is present in $[0,0.5]$, it is also present in $[0,1]$. Or $\{a,u\}\subset\{a,e,i,o,u\}$ means that the set on the right contains all letters from the set on the left (alphabet example).

Thus the whole issue is senseless, we are not allowed to write $A\in A$ since we should have elements on the left hand side, not sets. For example, we cannot say that $\{a,e,i,o,u\}\in\{a,e,i,o,u\}$ because the elements of the set on the right hand side are letters, not sets, so we cannot say that another set is its element.

And if you wish to go further, in probability theory a concept of $\sigma$-algebras $\cal F$ is introduced. Like sets were collections of elements, $\sigma$-algebras $\cal F$ are collections of sets. In this case, sets of elements become elements themselves (but they are not of the same type as elementary elements). For these elements you can use the symbol $\in$ with a set on the left hand side and $\sigma$-algebra on the right hand side.

Answer 7

Element : all the things seperated by commas in between braces, are elements, period, that is it, no confusion, blindly follow this.

Set : brace open, few elements seperated by commas brace closed. If you see an order like this, it is a set.

Example - {1,2,3}, here, 1, 2 and 3 are the elements of the set.

{{1},{2},{3}}, here {1}, {2} and {3} are elements.(yes they are individually sets too.)

Now coming to the question,

If there is a SET A, {1,2,3} and SET B, {1,2,3,4}

Then A is a subset of B. Note that subset is only used to establish relation between DEFINED SETS or UNIVERSAL SETE. Just like I have pre defined the above sets A and B.

Next, 1 BELONGS TO A(and B too). Note that we use BELONGS TO establish relation between an ELEMENT and a DEFINED SET.

EXAMPLES -

Let A = {{1},{2},{3},{4}}

Then {1} BELONGS TO A {1} is not a subset of A, just because {1}, though a set, has not been predefined neither is it universal.

Summarizing, subsets are for defined or universal sets and belongs to is for an element and a defined of universal set.

I guess, that about resolves it.

P.s : Nowadays in textbooks or exams such hard and fast rules are not used and the situation is usually self-explanatory.