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Variables and Truth Sets

  1. Aug 25, 2011 #1
    1. The problem statement, all variables and given/known data

    What are the truth sets of the following statements? List a few elements of the truth set if you can.

    c) x is a real number and 5[itex]\in[/itex]{y[itex]\in[/itex]R|x[itex]^{2}[/itex]+y[itex]^{2}[/itex]<50}

    3. The attempt at a solution

    I believe this says 5 is a member of the set of possible values for y, while y is both a real number and satisfies the condition x^2 + y^2 < 50. Therefore, the set of possible values for x is just the real numbers -7 through 7, subject to the condition that x^2+y^2 < 50.

    This seems too simple. Am I missing something here?

    Alternatively, it may be saying that 5 is a real number and that it satisfies the condtion x^2+5^2 < 50, in which case x would be the set of real numbers between -sqrt(25) and +sqrt(25). I'm not sure which reading, if either, is correct.

    Thanks in advance.
     
  2. jcsd
  3. Aug 26, 2011 #2
    We have two statements here. First, we have the statement that "x is a real number." This can be either true or false. Second is the statement that 5 is an element of the set {y in R | x^2 + y^2 < 50}. Since the set does not specify what type of object x is, the truth of the second statement is going to depend on the truth value of the first statement.

    So, assume x is a real number and ask yourself if it is possible that 5 is an element of that set.

    Then, assume x is not a real number (that the statement "x is a real number" is false), and determine if 5 can still be in the set. This will require some assumptions about what x is, though.. Perhaps x is complex? Then you'll need to think about possible complex values of x that make the statement x^2 + y^2 < 50 make sense (since complex variables lack ordering).

    This is what I am getting from the question.
     
  4. Aug 26, 2011 #3

    Mark44

    Staff: Mentor

    It's given that x is a real number, so there is no need to assume that x could be complex. In "the truth sets of the following statements," the "statements" are to be applied to the various parts of the problem, of which we see only part c.

    The inequality x2 + y2 < 50 defines a geometric figure. The truth set of this problem is the set of points where x and y are real and y satisfies x2 + y2 < 50. The problem is asking you to describe this set and list a few points in the set.
     
  5. Aug 26, 2011 #4

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Where in the world did you get "7"? I know that [itex]7^2= 49< 50[/itex] but why limit yourself to integers? [itex]7.05^2= 49.7025< 50[/itex] also.
    If [itex]5\in \{y|x^2+ y^2< 50\}[/itex] then we certainly must have [itex]x^2+ 25< 50[/itex]. What does that tell you about the possible values of x?
     
  6. Aug 27, 2013 #5
    HallsofIvy is definitely right.

    Remember the initial part, x2+y2<50, not only do you have TWO variables, but 52+52CANNOT be <50

    So you essentially have the total set of:

    {x[itex]\in[/itex]R| 5[itex]\in[/itex]{y[itex]\in[/itex]R | x2+y2<50}

    What this is essentially saying that 5 is an element of y and is also a real number. So...

    {x[itex]\in[/itex]R| x2+52<50}
    {x[itex]\in[/itex]R| x2+25<50}

    Reduced...
    {x[itex]\in[/itex]R| -5>x<5}

    Which makes sense, as you can't have two 52 otherwise that would be 50, which would make the set false.
     
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