in Mathematical Logic
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5 votes
5 votes

How can sentence be translated into a logical expression ?

"You can't ride the roller coaster if you are under 4 feet tall unless you are older than 16 years old".

Can answer be (!s->r)->!q

Where

q= You can ride the roller coaster

r=You are under 4 feet tall

s= You are older than 16 years old

in Mathematical Logic
3.2k views

4 Comments

$p=\text{you can ride roller coaster}$

$q=\text{you are under 4 feet tall}$

$r=\text{you are older than 16 years}$

 

$(q \wedge r^{'}) \rightarrow p^{'}$
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p: you can ride the roller coaster.

q: you are under 4 feet tall.

r: you are old then 16 years.

[!p if q unless r]

actual meaning of this [!p if q and !r]

[q ^ !r] -> !p.

i think.
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a: you can ride the roller coaster.

b: you are under 4 feet tall.

c: you are old then 16 years.

we know that q unless -p is equal to p->q

-c->b lets take it x

and we know q if p        p->q

so x->-a

(-c->b)->-a
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4 Answers

3 votes
3 votes
Best answer

If x then y unless z
(x Λ ~z) --> y
"You can't ride the roller coaster if you are under 4 feet tall unless you are older than 16 years old".

q= You can ride the roller coaster
r=You are under 4 feet tall
s= You are older than 16 years old

(r Λ ~ s)--> ~q should be the answer!

selected by

4 Comments

Yes, corrected. Thanks! @joshi_nitish
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Thanks @ just_bhavana for  the explaination...:)
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welcome :)
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3 votes
3 votes
Let Q :you can ride roller coaster

R : you are under 4 ft tall

S: you are older than 16 years old

(R AND (NOT S))--> NOT (Q)

4 Comments

edited by
But doesn't unless means OR. So please explain how come AND is used here.
This example of Kenneth H. Rosen is confusing.
R→ ((NOT Q) V S)
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Covert English sentence into logical expression

 

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

 

My approach

p - You can ride the roller coaster

q - You are under 4 feet

r - You are older than 16 years old.

 

Rewriting the sentence

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

 

 

and we know q unless not p translates to p -> q.

so translating this

 

we get

q-> ( ~r -> ~p) which can be further simplified to

 

q -> ( r OR ~p)

 

~q OR ( r OR ~P)

 

(~q OR r ) OR ~P

 

~(q AND ~r) OR ~P

 

(q AND ~r) -> ~P.

 

Now this was the final result given Kenneth Rosen.

 

But I want to know how can we come to this conclusion directly as mentioned in the book. It directly states the answer.

Please explain.

Thank You.
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how you use implication can you please explain in some question we use implication in different side and some question we use different, please provide some detail about implication so that i easly  can decided
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1 vote
1 vote

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

Let $p,q,r$ be the following propositions :

$p:$ “You can ride the roller coaster.”

$q:$ “You are under 4 feet tall.”

$r:$ “ You are older than 16 years old.”

Now, Let's Express the sentence in terms of propositions $p, q,$ and $r$ :

$(q ∧ ¬r) → ¬p$

1 comment

i agree with you totally but somehow i am bit confused.

from rosen,

for $a\rightarrow b$,

$b\,\, \text{unless}\,\, a^{'}$

so taking everything,

$(r^{'} \rightarrow q) \rightarrow p^{'}$
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0 votes
0 votes

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"
  • (Q  OR R) -> P'
  • should n't this be answer here
edited by

4 Comments

I think it should be ¬P in your implication
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yes it should be p complement
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A∨B is A unless B [usually] and is A except when B [usually].

(Q AND R')->P' both ways are correct to represent unless
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In rosen it is given that P-->Q  means Q unless not P.

so in above example "Q unless R" is equivalent to (~R-->Q) which could be expanded as (~(~R) or Q)  or simply , (R or Q)

 

but I am unable to digest (Q and ~R).

Please help where I am lacking.
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