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Let $P(x,y)$ be the predicate $2x+y = xy$, where the domain of discourse for $x$ and $y$ is integers. Determine the truth value of each statement.

$P(-1,1)$ : true

$\exists xP(x,y)$ : true

$\exists yP(4,y)$ : false

$\forall yP(2,y)$ : false

$\forall x \exists y(x,y)$ : false

$\exists y \forall x(x,y)$ : false

Have I answered correctly or have not?

Please guide me if I'm wrong. thank you.

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  • $\begingroup$ Answers should include justification. Why do you think each is so? (But they do seem okay after a cursory check.) $\endgroup$ – Graham Kemp Sep 30 '16 at 10:27
  • $\begingroup$ I'm a student there for i need a teacher to testify my question's answer, are they right or wrong. $\endgroup$ – zuby Sep 30 '16 at 10:35
  • $\begingroup$ When asking for solution verification, please remember that we don't have answer sheets to compare answers. You'll get better results if show your working so we can verify the process without having to solve everything from scratch. $\endgroup$ – Graham Kemp Sep 30 '16 at 11:05
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Interpretation of third statement:

Expression $\exists y P(4,y)$ means: There is at least one $y$ such that $P(4,y)$ is satisfied. Check:

$2\cdot 4 +y=4y\\\\ -3y=-8\\\\ y=\dfrac{8}{3}$

So, third statement is false because only possible value for $y$ is a fraction.

Other statements are answered correct as well.

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    $\begingroup$ The domain is the Integers. $\endgroup$ – Graham Kemp Sep 30 '16 at 10:30
  • $\begingroup$ Oh, right. Then all the statements are correct. $\endgroup$ – MaliMish Sep 30 '16 at 10:32
  • $\begingroup$ Because the P(4,y) must be satisfied. If y=2 then equation 2*4+2=2*4 is false. y should be 8/3 to satisfy equation but that is not integer, it is fraction. $\endgroup$ – MaliMish Sep 30 '16 at 10:42
  • $\begingroup$ sorry i wrongly posted that comment...I understood your last comment thank you very much $\endgroup$ – zuby Sep 30 '16 at 10:45
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Indeed, you've answered correctly in all but the second statement.

$∃xP(x,y)$ corresponds to the statement "There exists an $x$ such that $2x+y = xy$. The truth-value is indeterminate, because we know nothing about $y$; it is unbound, whether such a y exists, etc. to make the statement true or false.

Suggestion: When you submit your answers, if space is available, show your work to justify your conclusions. If nothing else, record the statements, your conclusions, and the reasoning that led you to the conclusions; those notes will handy notes to have.

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