1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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]
    [/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]
    [/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


    User Avatar
    Science Advisor

    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


    User Avatar
    Science Advisor

    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]
    [/tex]B as it is fine to say A [tex]

    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


    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Education Advisor

    [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


    User Avatar
    Science Advisor

    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].
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook