Cfrgtkky

You cannot ride the roller coaster if you are under 4 feet tall unless you are older than 16 years old.

Let:

• $P$ stands for "you can ride the roller coaster"
  


• $Q$ stands for "you are under 4 feet tall"
  


• $R$ stands for "you are older than 16 years old"
  

Is this logical expression correctly translated?

$$P rightarrow (Q wedge R)$$

The suggestion of $Pto (Q wedge R)$ would say that in order to ride the roller coaster you must be at least $4$ feet tall and you must me at least $16$ years old. But I would say the meaning of the given sentence is that you need to satisfy one of the age and height conditions, not both.

I think the sentence means: In order to ride the roller coaster, you must be at least $4$ feet tall, or you must be over $16$ years old.

Symbolically (using your $P, Q, R$), this would be $Pto (Qvee R)$. In contrapositive form (which would tell you what keeps you from riding the roller coaster: $(neg Pwedge neg Q)to neg R$. (If you are under 4 feet tall and younger than $16$, then you can't ride the roller coaster).

(1) 'Unless' statements:

We have some strategies to transform 'unless' clauses in conditional statements. The most common seems to be directly translate it to a 'if not':

• I'm not coming to the party unless Sylvia comes.


• I wouldn't eat that food unless I was really hungry.

Where both respectively translate to:

• If Sylvia is not coming to the party, neither do I.


• If am not really hungry I wouldn't eat that food.

Optionally, we can still transform the above sentences in their contrapositive form:

• I am coming to the party if Sylvia does.


• I would eat that food If am really hungry.

(2) Your Answer:

Consider the English sentence

You cannot ride the roller coaster if you are under 4 feet tall unless you are older than 16 years old.

Note that its structure is the same as

If you are under 4 feet tall you cannot ride the roller coaster unless you are older than 16 years

Now we set up our glossary.

Let:

• $P$ means 'you $color{red}{can}$ ride the roller coaster'


• $Q$ means 'you are under 4 feet tall'


• $R$ means 'you are older than 16 years old'

Your answer is

$$ Q to (P to R)$$

(Note that we translated the 'unless' clause direclty to its contrapositive form)

Solution: let suppose
q= You can ride the roller coaster;
p= you are older than 16 years old;
r= you are under 4 feet tall;
There is two states of “q if p” and “q if r”;
Because
• q unless p :: (the statement is as )
You can ride the roller coaster unless you are not older than 16 years old;
• q, if r :: (the statement is as)
You can ride the roller coaster unless you are under 4 feet tall;

• These two statements have same conclusion that is “you can ride a roller coaster” and the hypothesis are two.
So these two conclusions may be simplify in one statement as:
“You can ride the roller coaster, if you are under 4 feet tall and you are not older than 16 years old”

Then in this statement there is the (q, if p) form of implication and p which is the hypothesis has the operator “and” .So, the expression is
Hypothesis -> conclusion as:
(r ^ ~p) -> ~q
Where ~p means: you are not older than 16 years old” and
~q means: You cannot ride the roller coaster.