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Which predicate logic means, "There are at most two men in class"?

$\forall x\forall y((Male(x)\wedge Male(y))\rightarrow (x=y \vee y=x)$

$\exists x \exists y (Male(x)\wedge Male(y) \wedge x\neq y \wedge \forall z(Male(z) \rightarrow (z=x \vee z=y)))$

$\forall x \forall y \forall z (Male(x) \wedge Male(y) \wedge (Male(z)) \rightarrow (x=y \wedge x=z \wedge y=z)$

 

How do I know the meaning of the statements? I want to know the working steps.

1 Answer

Best answer
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7 votes
$\forall x\forall y((Male(x)\wedge Male(y))\rightarrow (x=y \vee y=x)$

For every $x,y$, if both are male, then $x = y$ or $y = x$.

The second part of OR is redundant here as $x = y$ is same as $y = x$. So, this basically means there cannot be no more than 1 man in the class (either 0 or 1 man is possible).

$\exists x \exists y (Male(x)\wedge Male(y) \wedge x\neq y \wedge \forall z(Male(z) \rightarrow (z=x \vee z=y)))$

There are $x$ and $y$ such that both are men, and if there exist another man $z$, then $x = z$ or $y = z$. This means there are exactly two men in the class- no less no more.

$\forall x \forall y \forall z (Male(x) \wedge Male(y) \wedge (Male(z)) \rightarrow (x=y \wedge x=z \wedge y=z)$

For every $x,y,z$, if all are men, then all are the same. I guess $z$ is redundant here and it should mean the same as in (1).
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The answer is option c. I want to know how to solve these type of questions and how other options are not the correct answer.