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Relative complement or difference between sets

Sal shows an example finding the relative complement or difference of two sets A and B. Created by Sal Khan.

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  • leaf blue style avatar for user David Elijah de Siqueira Campos McLaughlin
    Could you add, divide and/or multiply sets?
    (96 votes)
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    • leaf green style avatar for user SteveSargentJr
      Great question! Actually, there are operations you can do with sets that are similar to the operations of multiplication, division, etc. that you do with numbers. However, that is all pretty advanced stuff--you probably won't learn about them until at least high school (if not college). Nevertheless, these operations do exist (and they have fancy names like Minkowski Addition, Product Sets, & Quotient Sets, to name a few).

      If you're still interested, here's an example of an operation on two sets to get you thinking about the concept:
      Let's say you have two sets, A & B
      A = {a, b, c}
      B = {2, 7}

      Then the Cartesian Product of A and B (written as "A x B") is the set:

      A x B = { (a,2), (a,7), (b,2), (b,7), (c,2), (c,7)},

      where each element of the set is an ordered pair of elements. Thus, while "7" is a single element of the set B, the letter/number pair "(c,2)" is a single element of the set "A x B". Also, note that while "(c,2)" is an element of "A x B", "(2,c)" is NOT an element of "A x B" because order matters.

      Anyway, there's a lot more to be said about these set operations but I don't want to bore you. I hope you found this interesting! (And not too confusing!!)
      (186 votes)
  • aqualine ultimate style avatar for user TheAwer
    EDIT: Can we even have the same object more than once in a set?
    At Sal says "take out the 17, 19, and 6s". So does that mean that if there were two 17s in set A, but only one in set B, would you take out both or only one 17? Do you remove all of what is in set A from set B, or only how many items are in set A from set B? (I can clarify my question if needed)
    Example:
    Set C={1, 2 , 2 ,5,12, 33 ,chicken, 33 }
    Set D={ 2 ,pizza, 33 }
    When you take C-D, is it (paying attention to the 2 s and 33 s)
    C-D={1, 2 ,5,12,chicken, 33 } (we subtracted one 2 & 33 from set C because there was only one 2 & 33 in set D)
    OR
    C-D={1,5,12,chicken} (we subtracted all 2 s & 33 s from set C because there is at least one 2 & 33 in set D)
    Which way is correct?

    Thanks!
    (24 votes)
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  • leaf green style avatar for user Robin Thomas
    As per the video, is there any difference between A-B and A-A∩B ?
    (11 votes)
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  • piceratops tree style avatar for user K.492
    Does it matter in what order species appear in sets?

    For example is A = {5, 24, 6, 7}
    the same as set B = {24, 7, 6, 5} or would you treat them as different sets?

    Thanks in advance!
    (3 votes)
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    • ohnoes default style avatar for user webuyanycar.com
      Yes, you must treat them as different sets. In this case, each set is given a different name. The first is A, the second is B. Even though the ORDER of the items in a set does not matter, the NAME does. So, by giving these sets two different names, you have created two different, distinct sets. You must treat them as such. Hope this helps!
      (2 votes)
  • piceratops ultimate style avatar for user shubhangshrivastava09
    If A-B = ø then A=B is this true or false
    (3 votes)
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  • male robot johnny style avatar for user Dandy Cheng
    Since A\B = {5, 3, 12}, can I use a notation like this?

    A'

    Would the notation above do the same thing? Or do I need a universal set in order to use that notation? I've learned that notation from my teacher.
    (2 votes)
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    • leaf blue style avatar for user Dr C
      The first notation means everything in A but not in B.
      The second notation means everything not in A.

      Regardless of a universal set, these are not the same thing.

      And you really can't not have a universal set. Even if it's not stated explicitly, a universal set is probably implied.
      (5 votes)
  • marcimus pink style avatar for user kaivalya.panyam
    PLEASE dont laugh at my ignorance............as far as i know SET IS A COLLECTION OF WELL DEFINED OBJECTS.... What is the well defined object in the null/empty set? How can we call a set an "EMPTY SET"?
    (2 votes)
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  • aqualine ultimate style avatar for user Priscilla
    At / before, does the 6 float off to space?
    (2 votes)
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  • purple pi purple style avatar for user Judah Hoover
    Null is different than zerro right?
    (0 votes)
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    • hopper jumping style avatar for user Paul Hodgson
      The symbol for a null set does look like a zero doesn't it? But I think that's where the similarities end as it would have been entirely possible for set A to contain a zero in the same way that it could have contained a badger.

      In this instance Null means that the set has no member items in it.
      (9 votes)
  • duskpin tree style avatar for user Mikaiah Oxford
    Is ø a subset of every set?
    (2 votes)
    Default Khan Academy avatar avatar for user

Video transcript

What we're now going to think about is finding the differences between sets. And the first way that we will denote this is we'll start with set A. I've already defined set A. Let me do it in that same shade of green. I've already defined set A here. And in both cases, I've defined these sets with numbers. Instead of having numbers as being the objects in the set, I could have had farm animals there or famous presidents, but numbers hopefully keep things fairly simple. So I'm going to start with set A. And from set A, I'm going to subtract set B. So this is one way of thinking about the difference between set A and set B. And when I've written it this way, this essentially says give me the set of all of the objects that are in A with the things that are in B taken out of that set. So let's think about what that means. So what's in set A with the things that are in B taken out? Well, that means-- let's take set A and take out a 17, a 19-- or take out the 17s, the 19s, and the 6s. So we're going to be left with-- we're going to have the 5. We're going to have the 3. We're not going to have the 17 because we subtracted out set B. 17 is in set B, so take out anything that is in set B. So you get the 5, the 3. See, the 12 is not in set B, so we can keep that in there. And then the 19 is in set B, so we're going to take out the 19 as well. And so that is this right over here is-- you could view it as set B subtracted from set A. So one way of thinking about it, like we just said, these are all of the elements that are in set A that are not in set B. Another way you could think about it is, these are all of the elements that are not in set B, but also in set A. So let me make it clear. You could view this as B subtracted from A. Or you could view this as the relative complement-- I always have trouble spelling things-- relative complement of set B in A. And we're going to talk a lot more about complements in the future. But the complement is the things that are not in B. And so this is saying, look, what are all of the things that are not in B-- so you could say what are all of the things not in B but are in A? So once again, if you said all of the things that aren't in B, then you're thinking about all of the numbers in the whole universe that aren't 17, 19, or 6. And actually, you could even think broader. You're not even just thinking about numbers. It could even be the color orange is not in set B, so that would be in the absolute complement of B. I don't see a zebra here in set B, so that would be its complement. But we're saying, what are the things that are not in B but are in A? And that would be the numbers 5, 3, and 12. Now, when we visualized this as B subtracted from A, you might be saying, hey, wait, look, look. OK. I could imagine you took the 17 out. You took the 19 out. But what about taking the 6 out? Shouldn't you have taken a 6 out? Or in traditional subtraction, maybe we would end up with a negative number or something. And when you subtract a set, if the set you're subtracting from does not have that element, then taking that element out of it doesn't change it. If I start with set A, and if I take all the 6s out of set A, it doesn't change it. There was no 6 to begin with. I could take all the zebras out of set A; it will not change it. Now, another way to denote the relative complement of set B in A or B subtracted from A, is the notation that I'm about to write. We could have written it this way. A and then we would have had this little figure like this. That looks eerily like a division sign, but this also means the difference between set A and B where we're talking about-- when we write it this way, we're talking about all the things in set A that are not in set B. Or the things in set B taken out of set A. Or the relative complement of B in A. Now, with that out of the way, let's think about things the other way around. What would B slash-- I'll just call it a slash right over here. What would B minus A be? So what would be B minus A? Which we could also write it as B minus A. What would this be equal to? Well, just going back, we could view this as all of the things in B with all of the things in A taken out of it. Or all of the things-- the complement of A that happens to be in B. So let's think of it as the set B with all of the things in A taken out of it. So if we start with set B, we have a 17. But a 17 is in set A, so we have to take the 17 out. Then we have a 19. But there's a 19 in set A, so we have to take the 19 out. Then we have a 6. Oh, well, we don't have to take a 6 out of B because the 6 is not in set A. So we're left with just the 6. So this would be just the set with a single element in it, set 6. Now let me ask another question. What would the relative complement of A in A be? Well, this is the same thing as A minus A. And this is literally saying, let's take set A and then take all of the things that are in set A out of it. Well, I start with the 5. Oh, but there's already a 5. There's a 5 in set A. So I have to take the 5 out. Well, there's a 3, but there's a 3 in set A, so I have to take a 3 out. So I'm going to take all of these things out. And so I'm just going to be left with the empty set, often called the null set. And sometimes the notation for that will look like this, the null set, the empty set. There's a set that has absolutely no objects in it.