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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.

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  • purple pi purple style avatar for user Nastya Safonova
    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)
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    • leaf green style avatar for user EdorFaus
      Like he mentions at , 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).
      (154 votes)
  • leaf orange style avatar for user Savannah Noel
    So subsets and supersets are the same thing?
    (12 votes)
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    • piceratops ultimate style avatar for user Kory Dondzila
      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.
      (31 votes)
  • purple pi purple style avatar for user Ryan Christopher
    Is it necessary to have the underline? if not, is the "equal to" property assumed without the underline?
    (5 votes)
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    • leaf blue style avatar for user Matthew Daly
      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.
      (18 votes)
  • blobby green style avatar for user Parshand Doctor
    is a null set a strict subset of itself
    (5 votes)
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    • leaf blue style avatar for user Dr C
      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.
      (12 votes)
  • blobby green style avatar for user Shubham OM Sinha
    if A ϵ B, B ⊂ C, then why not A ⊂ C ?
    (7 votes)
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  • starky ultimate style avatar for user Matthew Kim
    When would you use sets in real life?
    (6 votes)
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    • male robot hal style avatar for user Adam Dreaver
      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)
  • male robot hal style avatar for user ledaneps
    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)
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    • leaf blue style avatar for user Matthew Daly
      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)
  • male robot hal style avatar for user Ritchie Joseph
    So, isn't every set a subset and a "strict" subset of the Universal Set?
    (2 votes)
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  • hopper cool style avatar for user David
    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!
    (1 vote)
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    • hopper cool style avatar for user JPhilip
      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. :)
      (6 votes)
  • blobby green style avatar for user Anjan Kumar Das
    If a set contains 2 (as a example) elements , what will be the number of sets of it's proper subset ?
    (2 votes)
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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.