Pugh problem 1.3: quantified statements

83 Views Asked by At

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.