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Recast the following English sentences in mathematics, using correct mathematical grammar. Preserve their meaning.

a) $2$ is the smallest prime number.

b) The area of any bounded plane region is bisected by some line parallel to the $x$-axis.

c) "All that glitters is not gold."

After some help in the comments, I have a revised attempt.

a) Define the predicate $P(n)$ for $n$ is a prime number. Then: $$P(2) \wedge \forall p, \; P(p) \implies p \geq 2.$$ b) I am still unsure on how to write part (b).

c) Define the predicate:

Glitter($x$): $x$ glitters Gold($x$): $x$ is gold

The quantification: $$\forall x, \; \text{Glitter}(x) \implies \neg \text{Gold}(x).$$ The problem with (c) is that the universe of discourse is not specified, so when I say "for each $x$," it is not clear what I am referring to.

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John P.
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  • $\exists x=2$ is not syntactically correct. – Mauro ALLEGRANZA Mar 04 '20 at 08:09
  • The existential quantifier is not needed, using the term $2$. "2 is the smallest prime number" can be unwinded into: "2 is prime and every prime is greater-or-equal to it". $\text P(2) \land \forall p (\text P(p) \to p \ge 2)$. – Mauro ALLEGRANZA Mar 04 '20 at 08:11
  • For c) Yes; you have to use two predicates: Glitter(x) and Gold(x). – Mauro ALLEGRANZA Mar 04 '20 at 08:13
  • Thank you for the comments. I've revised my answer with another attempt. In (c) in particular, I am not sure what the domain of discourse is, so ir doesn't seem that we have defined $x$ in any way. – John P. Mar 04 '20 at 17:14
  • Re your new c): "All that glitters is not gold" is Ok. "All" means all, thus the domain is "everything". – Mauro ALLEGRANZA Mar 05 '20 at 12:49

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