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Is it if/then or and statement?

  1. Mar 22, 2015 #1
    Mentor note: moved to homework section

    I'm having trouble determining if statements are combined using if/then or and operators. Take the following statements:

    1) Nobody in the calculus class is smarter than everybody in the discrete math class.

    For 1 I'd think a proper way to term it is "For not any x's and all y's, if x students are taking calculus, and y students are taking discrete then the x students are smarter than the y students."

    With notation, it'd be -ExVy( C(x) and D(y) ---> S(x,y) ).

    (note that I use E to represent for some and V to represent for all)

    This answer agrees with one solutions manual but not with another. The other one says: ¬∃xC(x)∧[∀yD(y)∧S(x,y)] But wouldn't this literally mean that " no one is taking calculus and everyone is taking
    discrete and no calculus students are smarter than discrete math students"? That latter bit agrees with the statement, but not former, which I believe says "nobody takes calculus"



    2) Everyone likes Mary, except Mary herself.

    I thought this looked like a fairly straightforward "and" statement. " For all x's,All people x like Mary AND x is not equal to M"

    Symbollically," Vx L(x,m) and m not = to x"

    But according to the answers manual, the answer is: Vx ( -(x=m) --> L(x,m)). Wouldn't this mean if Mary is not equal to x, then everyone likes Mary? It's somewhat similar but my answer seems a bit more straightforward. Is it possible that there are just many ways to write one statement?
     
    Last edited by a moderator: Mar 22, 2015
  2. jcsd
  3. Mar 22, 2015 #2

    Mark44

    Staff: Mentor

    You need to define what x and y mean.
    All of the correct notation is available here. Click the ##\Sigma## symbol at the upper right end of the menu bar to see what is available.
    Or - ∨
    And - ∧
    There exists - ∃
    For all - ∃
    Element of - ∈
    Not an element of -∉
    Implies - ⇒

    Apparently we don't have a symbol for "Not"
    Again, you need to define your symbols. What does C(x) mean? I assume it has something to do with calculus students, and D(y) has something to do with discrete math students, but you should make clear what these mean.

    What does S(x, y) mean?

    For what it's worth, the sentence you're working with is equivalent to, "Everyone in the calculus class is dumber than any student in the discrete math class."
     
  4. Mar 22, 2015 #3

    Yes, as you said. C(x) are all kids taking calculus and D(y) are all kids taking discrete. S(x,y) refers to calculus kids being smarter than Discrete kids.
     
  5. Mar 22, 2015 #4

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    To start with, before worrying about "if then" or "and", you are interpreting the statement incorrectly!

    "Nobody in the calculus class is smarter than everybody in the discrete math class." Your "For not any x's and all y's, if x students are taking calculus, and y students are taking discrete then the x students are smarter than the y students" in addition to the awkwardness of "for not any x", you are essentially saying that every student in discrete math is smarter than every student in the Calculus class"- and that is NOT what the original sentence means. It means, rather, that for any student in the Calculus class there is at least one student in the discrete math class that is at least as smart. That is, "for all x in Calculus, there exist at least one y in discrete math such that y is as smart, or smarter, than x." Equivalently, taking [itex]\ge[/itex] to mean "is at least as smart as", "if x is in Calculus then there exist y in discrete math such that ##y\ge x##."

    Similarly, "Everyone likes Mary, except Mary herself" is NOT "For all x's,All people x like Mary AND x is not equal to M" because, as long as Mary exists there IS "x" such that is equal to Mary! Instead you want "For all x, either x likes Mary or x= Mary".
     
  6. Mar 22, 2015 #5

    Mark44

    Staff: Mentor

    OK, what does "C(x) and D(y)" represent? It would have to have a value of either true or false, wouldn't it?
     
  7. Mar 22, 2015 #6
    Thanks, this made perfect sense. Just to make sure, you're saying if I stated "for all x, All x's like Mary",pairing that up with "and Mary is not equal to x" would essentially be a contradiction ?
     
  8. Mar 22, 2015 #7

    vela

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    Your phrasing is a bit awkward. When you say "for all x", you shouldn't then say "All x's like Mary." The x stands for a single person, so you should say "x likes Mary" instead. "For all x, x likes Mary" would mean everyone likes Mary (including Mary herself).

    I think the answer you quoted from the book is wrong. It says that if x isn't Mary, then x likes Mary, but it doesn't say what's true if x is Mary. Mary could like herself or not. Either way, the implication is true. The original wording, however, says Mary doesn't like herself.

    What Halls said is different than the book's answer because he used the exclusive OR: either x likes Mary or x is Mary. If x=Mary, that statement is true only if x likes Mary is false, i.e., Mary doesn't like herself.
     
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