1. Using our logical constructions for \"exactly one\" and exactly two\" express

Question

1. Using our logical constructions for "exactly one" and exactly two" expressions, convert these English sentences to logic such that their (a) Exactly one man was first on the Moon. (b) Exactly two cupcakes have chocolate frosting meanings are preserved. 2. As discussed in class, "exactly two" statements can be viewed as the intersection of "at most two" and at least two." Convert these English sentences to logic such that their meanings are preserved. (a) At least two movies earned $100 million. (b) Joaquin's car can have at most two passengers.

Explanation / Answer

1 . a) Consider the sentence "Exactly one man was first on the Moon. "

The example of exactly one sentence can be given as , (y (x P(x)=x=y))

The given english sentence can be wriiten in first order logic ,

x.man(x) OnMoon(x) y.(man(y) OnMoon(y) x = y)

b) . Consider the sentence given " Exactly two cup cakes have chocolate frosting. "

To write the exactly two sentence in First order logic , conjoin at least two and at most two sentences , The exactly two sentence can be given as, Exactly two sentence example is given as ,

x.y.P(x)P(y)

xy

(z.P(z)(y=xz=y)

The sentence can be given as ,

x, y.CupCakes(x) HaveChocolateFrosting(x) CupCakes(y) HaveChocolateFrosting(y) x y z. (CupCakes(z) HaveChocolateFrosting(z) (x = z y = z))

2. a) " At least two movies earned $100 million. "

First write the sentence as xy (movies(x) movies(y)). , where x and y have two different movies and to prove they are distinct objects use clause  x y.

x, y.movies(x) movies(y) earned$100million(x) earned$100million(y) x   y

b) . " Joaquin's car can have at most two passengers".

There are at most two passengers in Joaquin's car , It means that  if you make three draws from the car and get a passenger every time, then you must have drawn the same passenger more than once.

It can be given in first order logic as,

xyz ((Passenger(x) Passenger(y) Passenger(z)) (x = y y = z x = z)).

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