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I have to convert the given predicates to English sentences. I have tried from my side but I am very confused (as there can be many forms to represent a statement and also that my English statement might be wrong). Here are the predicates and English statements. Please tell me if there can be a better answer for any of these?

  1. $\forall x\exists y R(x,y)$

    For every $x$ there is at least one $y$ such that $R(x,y)$

  2. $\exists x\forall y R(x,y)$

    For every $x$ there is a single $y$ such that $R(x,y)$

    EDIT:

    There is an x $x$ for which there is a single $y$ such that $R(x,y)$

  3. $\forall x(\neg Qx)$

    For every $x$ $Q(x)$ is false; or $Q(x)$ is false for every $x$

  4. $\exists y(\neg P(y))$

    There is at least one $y$ for which $P(y)$ is false.

Please help me out. Thanks.

  • 0
    @Akito: No, you are misinterpreting it again. Number 2 says that there is at least one $x$ for which **every** $y$ satisfies $R(x,y)$. For example, if $R(x,y)$ meant "$x$ is less than or equal to $y$", then the statement would say "there is an $x$ which is less than or equal to each and every $y$". There is a *single* value of $x$ at issue.2011-09-22

1 Answers 1

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You have only mistake in number 2 as Arturo has pointed. The right expression is: There exists at least one $x$ such that for all $y$ the formula $R(x,y)$ is true. What you wrote is encoded as $\forall x\exists yR(x,y)$.

  • 0
    No need; your answer is good.2011-09-22