I'm preparing for a midterm and would like to ensure that I'm writing statements in quantified symbolic logic properly.
Since I'm only seeking someones expertise to check over my work, I can delete this afterwards since it's not really a question.
Let A(x) represent “x is high quality”, B(x) “x looks good”, C(x, y) “x is faster than y”, and D(x, y) “x is cheaper than y”. Translate the following statements into quantified symbolic logic, using the set of all cars S as the domain.
(a) All high quality cars look good.
- ∀x ∈ S, A(x) ^ B(x)
(b) There are no good-looking cars.
- ~∃x ∈ S, B(x)
(c) The faster a car is, the more expensive it is.
- ∃x,y ∈ S C(x,y) ⟹ D(x,y)
(d) There is a cheapest car.
- ∃x ∈ S, D(x,S) Would this work, I'm not too sure about it?
(e) Some good-looking cars are not high quality.
- ∃x ∈ S, B(x) ⟹ A(x)
Thank you very much! :)
a) is not right ... your statement says that every car is a high-quality car that looks good. ... use a conditional!
b) is correct
c) you need some operator between $C(x,y)$ and $D(x,y)$ ... also think about the order of the arguments: which is more expensive than which?
your d) is not grammatical : $S$ is used for the domain ... you can't use it inside a predicate ... try again
e) again you need to add some operator ... indeed, this statement clearly needs a negation somewhere ...