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Help with Set theory, compund statements

  1. Feb 12, 2015 #1
    The class is called Math for EE and CE. The professor teaches from his own notes and doesn't give many examples. Any help checking my work would be appreciated and/or if you could point me in the direction of more examples like these. I've looked trough Set Theory and discrete math books but nothing looks similar.

    1) Write out the following statement in English
    ∀a∈A: ∃b∈B: ∀c∈C : ((a+b>1 ∧ b-c >2) ⇒a+b+c >2)

    My answer:
    For all a∈A, all c∈C, there exists one b∈B, such that if a+b > 1 and b-c>2 is true then a+b+c > 2 is also true.

    2) Let A = [9] - [3] and B = {x∈A| x>4}
    Define by listing the set C = A - B

    My answer:

    A = [1,2,3,4,5,6,7,8,9] - [1,2,3]
    A= [4,5,6,7,8,9]

    B= [5,6,7,8,9]

    C= A - B = [4,5,6,7,8,9] - [5,6,7,8,9]
    C = [4]
     
  2. jcsd
  3. Feb 12, 2015 #2
    Be careful with your quantifiers. Changing the order of quantifiers does make a difference.

    For example: For all x in R, there exists a y in R such that y>x. This is a true statement, however,
    There exists a y in R such that for all x in R, y>x. This is clearly not a true statement.
     
  4. Feb 12, 2015 #3
    Sorry I'm not seeing the difference. I could see how it would be wrong if it were, there exists a x in R such that for all y in R, y>x. But the statements above seem the same.
     
  5. Feb 12, 2015 #4
    The first statement says: For every x, I can find a y that is larger.

    The second statement says: That you can find a single y that is greater than EVERY x. That is equivalent to claiming there is a largest real number. Your last reply was a claim that there is a smallest number.


    Maybe this is more clear:
    "For all" can also be read as "For every" or "For any"

    In the first example, it is might be more appropriate to read the statement as:
    For any x there is a y such that y>x.

    In the second, we might read it as:
    There is a y such that for every x, y>x.
     
    Last edited: Feb 12, 2015
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