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Homework Help: Discrete Math; Subsets

  1. Aug 24, 2010 #1
    1. The problem statement, all variables and given/known data

    Let their be a set A, and let B be the set: {A, {A}} (the set containing the elements A and the
    set that contains element A)

    As you know, A is an element of B and {A} is also an element of B.

    Also, {A} is a subset of B and {{A}} is also a subset of B.

    However, A is not a subset of B

    2. Relevant equations

    [URL]http://65.98.41.146/~carlodm/phys/123.png[/URL]


    3. The attempt at a solution

    See my drawing above. I created a Venn Diagram to deduce the logic with no clear results. In the first diagram to the left, I can clearly see that the element A is a subset of B ... yet.. it is not? Can someone explain?
     
    Last edited by a moderator: Apr 25, 2017
  2. jcsd
  3. Aug 24, 2010 #2

    Office_Shredder

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    I don't think Venn diagrams are the way to go. Let's do an example. Let A be the set of integers. We know that B has two elements: A and {A}. But A has 3 as an element. 3 is neither the set of integers or the set containing the set of integers, so 3 is not in B.

    In the venn diagram you are getting A as an object and A as a set containing other objects confused
     
  4. Aug 24, 2010 #3
    My question:

    Why are we comparing set A (the set of all integers) on an element-by-element basis? I agree that "3" is an element of A such that A = { ...,-1,0,1,0,1,2,3...}.

    I agree that "3" is neither the set of integers A = { ...,-1,0,1,0,1,2,3...} nor the set containing the set of integers {A} = {{ ...,-1,0,1,0,1,2,3...}} .

    I fail to see how individual elements of A are pertinent to the problem. If A is a subset of B, every element of A is in B.

    ----------------------------------------------------- Light Bulb in My Head

    After writing the last line of my argument, I realized that simple truth:

    "A is a subset of B iff every element in A is in B"

    Given your argument, "3" is an element of A yet 3 is neither the set of integers NOR the set containing the set of integers.

    Eureka moment. Thanks!
     
  5. Aug 24, 2010 #4
    Going a mile further...

    {A} is a subset of B in this example because....

    Every element in {A} is in B.

    The only element of the set of sets is A. A is the set of all integers. {A} therefore is the subset of B.

    ------------------------------------------------------------
    {{A}} is a subset of B since:

    Every element in {{A}} is in B.

    The only element in the set of sets of sets is A. The sets of sets of A, {{A}} is an element of B.

    That was correct, no?

    {{{A}}} is not a subset of B.... well because {{A}} is not in B?
     
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