Determine the propositional form and truth value of the following

Question

In "A Transition to Advanced Mathematics", the 8th edition, problem (7e), I must determine the propositional form and truth value for the following:

Although $51$ divides $153$, it is neither prime nor a divisor of $409$

Attempt

Here is my attempt:

If $P=\text{51 divides 153}$, $Q=\text{51 is prime}$, and $R=\text{51 is a divisor of 409}$, then the propositional statement can be described as the following:

$$P\land\sim(Q\lor R)$$

Also; since $P$ is true, $Q$ is false, and $R$ is false; the propositional statement has the following truth value:

\begin{align} & T\land\sim(F\lor F)\\ & T\land \sim F \\ & T\land T \\ & T \end{align}

Therefore the truth value of the propositional statement is true.

Question: Am I correct?

Book Details :
"A Transition to Advanced Mathematics 7th ED"
"A Transition to Advanced Mathematics 8th ED"

Authors :: Douglas Smith & Maurice Eggen & Richard St Andre

Answer 1

You are essentially Correct , & your Conclusion is Valid.

There are a few finer Points to high-light , which will aid in the more general Case.

(1) Neither Nor :

There are 2 ways to convert "neither X nor Y" :
(1A) "not X AND not Y" ...
(1B) ... which is Equivalent to "not (X OR Y)" by the Laws of DeMorgan.

Here (1A) is Direct , Easy & Smooth , not requiring Brackets , unlike (1B) which requires a way to negate a Bracket , which is generally unusual in Natural Languages.

OP Statement given is thus $ P \land (∼Q \land ∼R) $ , which is Equivalent to $ P \land ∼ (Q \lor R) $ , by the Laws of DeMorgan.

Supporting the (1A) View :

https://www.csm.ornl.gov/~sheldon/ds/ex1.1.2.html :

... the phrase "neither A nor B" is translated as "not A and not B". Additionally ...

https://stackoverflow.com/questions/5201034/translating-neither-nor-into-a-mathematical-logical-expression , Accepted Answer :

"Neither P nor Q" can be rephrased as "It is not the case that P, and it is not the case that Q"

https://www.cs.miami.edu/home/geoff/Courses/TPTPSYS/Practicum/EnglishToLogic.shtml , Point 6 :

Neither, nor. "Neither p nor q" means that both p and q are false. Therefore translate it ~p & ~q or ~(p | q). These two formulas are equivalent by DeMorgan's Theorem.

There are more ...

(2) Although :

When using Mathematical logic , "Conjunction" is almost always "AND" , while the English Language (& other Natural Languages) will have Conventional Customary Alternatives like :

but , yet , although , though , furthermore ...

There are more alternatives on Page 94 https://courses.umass.edu/phil110-gmh/text/c04.pdf , & there are other online articles listing the alternatives.

Mathematical logic will consider these alternatives all Equivalent to AND.
Humans will treat them a little Differently , due to Emotional Content & nuance in meaning , which is not necessary in Mathematical logic.
These alternatives are used when we want convey contrast & surprise in general.

The Authors might be using "Although" here to high-light that it has no additional meaning in Mathematical logic.
The readers should realize that it will not matter & there will be no change in the logical formulation.

These alternatives are also used when we want Proof By Contradiction like this :

We assumed $X$ & then showed that $Y \implies Z$ , now , we showed that $Z$ is not true Although $Y$ is true.
Hence $X$ must not be true.