2

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?

$$P \rightarrow (Q \wedge R)$$

user189029
  • 39
  • 1
  • 1
  • 5
  • 1
    $ (q \land \lnot r) \Rightarrow p $ – gamma Nov 03 '14 at 05:20
  • You can pretty much see that your answer must be wrong because it translates as "if you cannot ride the roller coaster then. . .", whereas the given statement says something like "if. . . then you cannot rider the roller coaster." In other words you have made the converse error. – David Nov 03 '14 at 05:23
  • $P$ must stand for 'you can ride the roller coaster'.. – Bruno Bentzen Nov 03 '14 at 11:04
  • As you can see from your source : Kenneth Rosen, Discrete mathematics and its applications (7th ed), page 17, the answer is : $(q ∧¬r) → ¬p$. – Mauro ALLEGRANZA Nov 06 '14 at 13:19

4 Answers4

4

The suggestion of $P\to (Q \wedge R)$ 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 $P\to (Q\vee R)$. In contrapositive form (which would tell you what keeps you from riding the roller coaster: $(\neg P\wedge \neg Q)\to \neg R$. (If you are under 4 feet tall and younger than $16$, then you can't ride the roller coaster).

paw88789
  • 41,907
3

(1) 'Unless' statements:

There are some known strategies to transform 'unless' clauses into conditional statements. The most common one seems to be directly translate them using 'if not':

  • I'm not coming to the party unless Sylvia comes.
  • I wouldn't eat that food unless I was really hungry.

The examples above can be respectively translated as follows:

  • If Sylvia is not coming to the party, neither am I.
  • If am not really hungry I wouldn't eat that food.

Alternatively, we can use their (reverse) contrapositive forms:

  • I am coming to the party if Sylvia is.
  • I would eat that food If was 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.

Following the above reasoning we have:

If you are not older than 16 years, then you cannot ride the roller coaster if you are under 4 feet tall.

which is the same as:

If you are not older than 16 years, then if you are under 4 feet tall you cannot ride the roller coaster.

Now let:

  • $P$ stand for 'you can ride the roller coaster'
  • $Q$ stand for 'you are under 4 feet tall'
  • $R$ stand for 'you are older than 16 years old'

The answer you are looking for is

$$ \neg R \to (Q \to \neg P).$$

Bruno Bentzen
  • 3,392
0

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.

noor fatima
  • 1
0

No, because $P\implies Q\land R$ actually means that You must be under 4 feet tall and older than 16 to ride the roller coaster which is not true. You can be under 4 feet tall and more than 16 years old and still ride the roller coaster. The sentence of the question is equivalent to:

If you are more than 16 years old or more than 4 feet tall or both, you can ride the roller coaster.

which is $$(\lnot Q\lor R)\implies P$$so, the if-else-then statements always work.

Another way to tackle these sort of problem is the truth table:

$$ \begin{matrix} Q&R&P \\0&0&1 \\0&1&1 \\1&0&0 \\1&1&1 \end{matrix} $$

Mostafa Ayaz
  • 33,056