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Thread: 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 . . .


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

    Determine the truth value of the statements.

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


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


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



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


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


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

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



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

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

    If $\displaystyle x = 5$, we have: .$\displaystyle 5 + 2y \:=
    \;5y\quad\Rightarrow\quad y \:=\:\frac{5}{4}$ ... not an integer.

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