Conversion of statements to math

[Mr Davis 97]

Homework Statement

1) 2 is the smallest prime number
2) The area of any bounded plane region is bisected by some line parallel to the x-axis.
3) All that glitters is not gold

Homework Equations

The Attempt at a Solution

1) $\forall p \in P ~~~ (2 \le p)$ (where $p$ denotes the set of prime numbers)

2) Let $r = \text{any bounded plane region}$, $l = \text{any line parallel to the x-axis}$.
$\forall r \exists l ~~~ (l ~\text{bisects}~ r)$

3) Let $g = \text{any earthly object}$
$\exists g ~~~ (\text{g glitters} \wedge \neg (\text{g is gold}))$

Is this at all right? I just kind of winged it, and I am assuming that there are a lot of better ways that these could be done.

 

[andrewkirk]

(3) is fine

(1) says that 2 is a lower bound for the prime numbers, but not that 2 is prime. You need to add a conjunct to specify that 2 is prime.

For (2) I imagine they want you to include the properties 'is a bounded plane region' and 'is parallel to the x axis' in the mathematical statement, rather than in the 'let' part. The verbal version of what I imagine is sought is something like:

For all r, if r is a bounded plane region then there exists l such that l is a line and l is parallel to the x-axis and l bisects the area of r.

 

[Mr Davis 97]

For 1) do you mean that I should write $\forall p \in P ~~~ (2 \le p) \wedge 2 ~~ \text{is a prime number}$

For 2) $\forall r ~~~ (r ~~ \text{is bounded by a plane region} \rightarrow \exists l ~~~\text{l is a line} ~~ \wedge ~~ \text{l is parallel to the x-axis} ~~ \wedge ~~ \text{l bisects the area of r})$. Is this the best that we can do given that there isn't a lot of notation for things like "parallel to" and "bisects?"

 

[andrewkirk]

For (1) you can write the second conjunct more concisely as $2\in P$, given that you defined $P$ as being the set of prime numbers.

(2) looks fine. There is no exact answer here because, to be exact, they'd need to tell us exactly what definitions of predicates and functions we should assume are available to us. Given they have not done that, we are free to assume the existence of any reasonable-sounding predicates we like, such as '#1 is parallel to the x axis' (a unary predicate) and '#1 bisects the area of #2' (a binary predicate).

I'd guess the general idea is to get the student accustomed to writing natural language sentences in logical form, using operators for things like conjunction, entailment and quantification, and employing predicates and functions where required.