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Confused about sets

  1. Jan 27, 2010 #1
    If A [tex]\subseteq[/tex] B does that mean A = B which means B = A because if A is a proper [tex]\subset[/tex] of B then A does not equal B right. I am wrong right?
     
  2. jcsd
  3. Jan 27, 2010 #2
    A [tex]
    \subseteq
    [/tex]B means that A is a subset of B. A could possibly equal B, but not in general.
    For example, if A = {1} and B = {1, 2, 3} then A is clearly a subset of B as all elements of A are also elements of B. Here, A is a proper subset of B.

    Usually, the symbol for a proper subset has a 'slash' through the horizontal line in the symbol [tex]
    \subseteq
    [/tex]. I can't seem to find it, however.

    It is useful to note that some people use the symbols [tex]\subseteq[/tex] and [tex]\subset[/tex] to mean the same thing.
     
  4. Jan 27, 2010 #3
    Yes exactly so if A [tex]\subseteq[/tex] B then every element in A must be in B and if A does not equal B then A is a proper [tex]\subset[/tex] of B.
     
  5. Jan 27, 2010 #4
    Yup, that is exactly what that means.

    Yes, if A[tex]\subseteq[/tex] B and A does not equal B, then A is a proper subset of B.
     
  6. Jan 27, 2010 #5
    But everytime A [tex]\subseteq[/tex] B that must mean A = B right? If not please give me an example. Thanks.
     
  7. Jan 27, 2010 #6

    sylas

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    The example was given to you in [post=2551076]msg #2[/post]. A = {1}, B = {1,2,3}.
     
  8. Jan 27, 2010 #7
    Right so from that example it would be wrong to say that A [tex]\subseteq[/tex] B but rather we should say A is a proper [tex]\subset[/tex] of B because A [tex]\neq[/tex] B.
     
  9. Jan 27, 2010 #8

    sylas

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    No. It is completely correct to say [itex] A \subseteq B [/itex].

    In the same way, it is completely correct to say [itex]3 \leq 5[/itex].
     
  10. Jan 27, 2010 #9
    Yes, in the example from message 2, A is a proper subset of B.
    However, it is fine to say A [tex]
    \subseteq
    [/tex]B as it is fine to say A [tex]
    \subset
    [/tex]B.

    The use of these two symbols are a matter of preference. Some professors will prefer to use one over the other but they both mean the same thing.

    In the link http://en.wikipedia.org/wiki/Naive_set_theory#Subsets, the notation for proper subsets is in the last line of the second paragraph.
     
  11. Jan 27, 2010 #10
    Okay I kind of get it. Thanks!
     
  12. Jan 27, 2010 #11

    vela

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    [itex]A \subseteq B[/itex] means "A is a subset of B"; [itex]A \subset B[/itex] means "A is a subset of B and A is not equal to B." If [itex]A=B[/itex], it would be accurate to say [itex]A \subseteq B[/itex] but not [itex]A \subset B[/itex]. If [itex]A\ne B[/itex] and A is a subset of B, either would be fine.
     
  13. Jan 27, 2010 #12

    Landau

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    As sylas mentioned, this is analoguous to [tex]<[/tex] and [tex]\leq[/tex]:

    [tex]x\leq y[/tex] means "[tex]x<y[/tex] or [tex]x=y[/tex]".

    [tex]A\subseteq B[/tex] means "[tex]A\subset B[/tex] or [tex]A=B[/tex]".

    (To deepen the analogy, they both define a partial order.)

    Of course, with this explanation you have to know that it is implicit in [tex]A\subset B[/tex] that A does not equal B.
     
  14. Jan 27, 2010 #13
    I think you just need to check their respective definitions. [tex]A \subseteq B[/tex] just means that [tex]\forall x\in A, x\in B[/tex]. This definition does not say anything about the elements in [tex]B[/tex]. In other words, [tex]\forall x\in B[/tex], it could be either in [tex]A[/tex] or not in [tex]A[/tex]. If [tex]\forall x\in B[/tex], implies [tex]x\in A[/tex], then [tex]A=B[/tex]; if not, then [tex]A\not=B[/tex].

    The definition of [tex]\subset[/tex] is that [tex]\forall x\in A[/tex], [tex]x\in B[/tex], and [tex]\exists y\in B[/tex], such that [tex]y\not\in A[/tex]. From this definition, we can see that actually, [tex]\subset[/tex] is a special case of [tex]\subseteq[/tex].
     
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