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Math Help - Truth value of the statement

  1. #1
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    Truth value of the statement

    P(x,y) means "x + 2y = xy", where x and y are integers.
    Determine the truth value of the statement.

    (a) P(1,-1).
    (b) P(0, 0).
    (c) ∃y P(3, y).
    (d) ∀x ∃y P(x, y).
    (e) ∃x ∀y P(x, y).
    (f) ∀y ∃x P(x, y).
    (g) ∃y ∀x P(x, y).
    (h) CS173 Discrete Mathematical Structures CS173 Discrete Mathematical Structures ∀x CS173 Discrete Mathematical Structures ∃y CS173 Discrete Mathematical Structures P(x, y).
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  2. #2
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    Hello, captainjapan!

    Your quantifiers didn't show up.
    I'll have to make some guesses . . .


    P(x,y) means " x + 2y \:=\: xy", where x and y are integers.

    Determine the truth value of the statements.

    (a)\;P(1,-1)
    We have: .  (1) + 2(-1) \:=\:(1)(-1) . . . True


    (b)\;P(0, 0)
    We have: . (0) + 2(0) \:=\:(0)(0) . . . True


    (c)\;\exists y\: P(3, y)
    We have:
    . . There exists y such that: . (3) + 2(y) \:=\:(3)(y) . . . True



    (d)\;\forall x\:\forall y\:P(x, y)
    This says: x + 2y \:=\:xy for all x and y . . . not true.
    [Find your own counterexample.]


    (e)\;\exists x\:\exists y\:P(x, y)
    There is an x and y such that: x + 2y \:=\:xy . . . True
    [See parts (a) and (b).]


    (f)\;\forall y\:\exists x\: P(x, y)
    This says: for any y, there is an integer x such that x + 2y \:=\:xy . . . not true.

    If y = 4, we have: . x + 8 \:=\:4x\quad\Rightarrow\quad x = \frac{8}{3} ... not an integer.



    (g)\;\exists y\:\forall x\:P(x, y)

    This says: for any x, there is an integer y such that x + 2y \:=\:xy . . . not true.

    If x = 5, we have: . 5 + 2y \:=<br />
\;5y\quad\Rightarrow\quad y \:=\:\frac{5}{4} ... not an integer.

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