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?
$$P \rightarrow (Q \wedge R)$$
$\endgroup$ 44 Answers
$\begingroup$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).
$\endgroup$ $\begingroup$(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
$$ \neg R \to (Q \to \neg P).$$
$\endgroup$ 2 $\begingroup$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.
$\endgroup$ 1 $\begingroup$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:
$$ \begin{matrix} Q&R&P \\0&0&1 \\0&1&1 \\1&0&0 \\1&1&1 \end{matrix} $$
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