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## Statistics and probability

### Course: Statistics and probability > Unit 7

Lesson 3: Basic set operations# Subset, strict subset, and superset

A subset of a set A is any set B such that every element of B is also an element of A. A strict subset is a subset that isn't equal to the original set (i.e. B must have at least one fewer element than A). A superset of A is any set C such that A is a subset of C. Created by Sal Khan.

## Want to join the conversation?

- What's the difference between a 'subset' and a 'strict subset'? If there is a strict subset than a 'not-strict' subset should exist as well. Could you please give an example of such difference?(46 votes)
- Like he mentions at1:20, the difference is that a
*strict*subset cannot be the same set, that is, it cannot contain all of the elements that the other set does. Or in other words, a*strict subset*must be smaller, while a*subset*can be the same size.

As an example, if A = {4,7} and B = {7,4} then A is a subset of B (because B contains all of the elements A does), but A is not a*strict*subset of B, because both sets contain exactly the same elements (B does not contain any element that A does not have).

However, if C = {7,4,1} then A is both a*subset*of C, and a*strict subset*of C, because C contains all the numbers A does*and also*a number A does not contain (the number 1).(150 votes)

- So subsets and supersets are the same thing?(1 vote)
- No, subsets have all their elements in another set while supersets contain all the elements of another set, though they may have more elements. Think of it this way...

A { 1,2,3,4 } B { 2,4 } B is a subset of A because all of its elements exist in A, A is not a subset of B because not all of its elements exist in B, but A is a superset because it contains all the elements of B and B is not a superset of A because B doesn't contain all of the elements of A, just 2.(30 votes)

- Is it necessary to have the underline? if not, is the "equal to" property assumed without the underline?(5 votes)
- Unfortunately, different mathematicians define these symbols in slightly different ways. Some say A⊂B to mean that A is a subset of B and A⊊B to mean that A is a proper subset of B. Other mathematicians say A⊆B to mean that A is a subset of B and A⊂B to mean that A is a proper subset of B. In theory, some mathematicians could also do like Sal and say A⊆B to mean that A is a subset of B and A⊊B to mean that A is a proper subset of B, although I don't actually recall ever seeing that in a textbook or paper. Mostly, it depends on the style of the mathematician and whether subsets or proper subsets are the more important concept in whatever branch of math you're reading about.(17 votes)

- is a null set a strict subset of itself(5 votes)
- No. For two sets A and B, the proper subset relation A ⊂ B implies that B contains at least one element which is not contained within A.

Denoting the null set with ∅, the statement A ⊂ ∅ would imply that ∅ contains at least one element which is not in A. However, the null set contains no elements, so the statement is impossible. There cannot be a proper subset of ∅. The null set is a subset of itself, but not a proper (strict) subset of itself.(11 votes)

- if A ϵ B, B ⊂ C, then why not A ⊂ C ?(7 votes)
- Think of this:

if A = { 1, 2, 3 }

then A ⊂ A and A ∉ A.

But if A = { A,1, 2, 3 }

then A ⊂ A and A ∈ A.

------

So to answer your question, unless A contains itself, it's not a subset of B or C.(5 votes)

- When would you use sets in real life?(6 votes)
- There are a few answers, but one answer is that more advanced math needs a rigorous foundation and often uses theorems and results that depend on set theory.

Advanced math can be so abstract or complicated that you must make sure your foundations are logically sound with no room for guess work or intuitive definitions.

The set of real numbers which applied maths uses everyday is constructed from the union of the sets of natural, rational, irrational and transcendental numbers.

One way to start this whole construction is starting with the construction of the natural numbers. One way to construct these natural numbers is using set theory! More information about this construction can be found at: http://en.wikipedia.org/wiki/Natural_number#A_standard_construction

So set theory allows you construct the foundations of mathematics and use advanced math that needs a rigorously constructed foundation.(4 votes)

- I was taught that the symbol for “is a member of,” or what we called “is an element of,” ("E" for "Element") is the lowercase Greek letter epsilon, or ε. Sal says that it is not. I’m OK with that. But here is something that I just noticed:

1. The "subset" symbol is a “C-shaped” symbol with a line**below**it.

2. The “element of” symbol is a “C-shaped” symbol with a line**through**it.

3. Continuing that same logic, if we keep the line rising, we would get a “C-shaped” symbol with a line**above**it.

Here’s my question:

Does such a symbol exist, and if it*does*, what does it*mean*?

I might have gone with “superset,” but apparently, superset is represented by the subset symbol written**backwards**.*(I can’t find a way to type the symbol I want, so I may have actually answered my own question.)*But please think this through with me anyway.(4 votes)- You're not totally wrong on the set membership notation. I am given to understand that it was originally a lowercase epsilon. But when set theory became the foundation for all of modern mathematics, a Greek letter was too important a symbol to give up. So the set membership symbol was tinkered with so that you could say "Let ε∈ℙ be given" without any fear of misunderstanding.

You're also right that there seems to be no symbol in Unicode for a subset symbol with a bar over it. If you ever wrote a paper that used that symbol it would mean whatever you defined it to mean, at least in that paper.(2 votes)

- So, isn't every set a subset and a "strict" subset of the Universal Set?(2 votes)
- Not all of them. The Universal set is not a "strict" subset of itself.(5 votes)

- Say there are 2 sets- A and B.

A is {2, 5, 8}

and B is {2, 5, 8}

What are all the relationships?

Are A and B just subsets of each other?

Thanks!(2 votes)- A = B, so:

A would be a subset of B,

B would be a subset of A,

A would be a superset of B,

and B would be a superset of A.

However,

A cannot be a strict subset of B,

B cannot be a strict subset of A,

A cannot be a strict superset of B,

and B cannot be a strict superset of A.

Hopefully this helps you. Tell me if I didn't properly answer your question, or if you still need help. :)(3 votes)

- If a set contains 2 (as a example) elements , what will be the number of sets of it's proper subset ?(2 votes)
- If a set contains 2 elements then there exist 3 sets that are proper subsets of the set.

For example the proper subsets of the set {1, 2} are: {1}, {2}, and {}.(3 votes)

## Video transcript

Let's define
ourselves some sets. So let's say the set A is
composed of the numbers 1. 3. 5, 7, and 18. Let's say that
the set B-- let me do this in a different
color-- let's say that the set B is
composed of 1, 7, and 18. And let's say that the set C is
composed of 18, 7, 1, and 19. Now what I want to start
thinking about in this video is the notion of a subset. So the first question
is, is B a subset of A? And there you might say,
well, what does subset mean? Well, you're a subset if
every member of your set is also a member
of the other set. So we actually can write
that B is a subset-- and this is a notation
right over here, this is a subset-- B is a
subset of A. B is a subset. So let me write that down. B is subset of A. Every
element in B is a member of A. Now we can go even further. We can say that B is
a strict subset of A, because B is a subset
of A, but it does not equal A, which means that there
are things in A that are not in B. So we could
even go further and we could say
that B is a strict or sometimes said a
proper subset of A. And the way you do that
is, you could almost imagine that this is kind of
a less than or equal sign, and then you kind of
cross out this equal part of the less than or equal sign. So this means a
strict subset, which means everything that
is in B is a member A, but everything that's in
A is not a member of B. So let me write this. This is B. B is a
strict or proper subset. So, for example, we can write
that A is a subset of A. In fact, every set is
a subset of itself, because every one of its
members is a member of A. We cannot write that A
is a strict subset of A. This right over here is false. So let's give ourselves a
little bit more practice. Can we write that
B is a subset of C? Well, let's see. C contains a 1, it contains
a 7, it contains an 18. So every member of
B is indeed a member C. So this right
over here is true. Now, can we write
that C is a subset? Can we write that
C is a subset of A? Can we write C is a subset of A? Let's see. Every element of C needs
to be in A. So A has an 18, it has a 7, it has a 1. But it does not have a 19. So once again, this
right over here is false. Now we could have
also added-- we could write B is a subset
of C. Or we could even write that B is a
strict subset of C. Now, we could also reverse
the way we write this. And then we're really just
talking about supersets. So we could reverse
this notation, and we could say that
A is a superset of B, and this is just another way of
saying that B is a subset of A. But the way you could
think about this is, A contains every
element that is in B. And it might contain more. It might contain
exactly every element. So you can kind of view
this as you kind of have the equals symbol there. If you were to view this
as greater than or equal. They're note quite
exactly the same thing. But we know already
that we could also write that A is a strict
superset of B, which means that A contains
everything B has and then some. A is not equivalent to B. So
hopefully this familiarizes you with the notions of subsets and
supersets and strict subsets.