1a) Let p, q, r be the following propositions: p = Bob is a statistician, q = Bo

Question

1a) Let p, q, r be the following propositions: p = Bob is a statistician, q = Bob likes numbers, r = Bob is fun at parties Translate each of the following sentences into a logical expression involving p, q, and r.

a. Bob likes numbers and he is a statistician.

b. Although Bob likes numbers and he is a statistician, he is also fun at parties.

c. Bob likes numbers but he is not a statistician.

d. For Bob to be a statistician, it is necessary for him to like numbers and be boring at parties.

e. Bob is a statistician if and only if he likes numbers and is fun at parties.

1b) Consider a group of climbers that climb together on weekends. Let the predicate C(x) denote that x has a climbing certificate, and let A(x, y) denote that climber x has assisted climber y. Translate each of the following statements into a logical expression involving C(x), A(x, y) and quantifiers. Let the universal set U be the set of climbers attending the monthly hikes.

a. Sylvia has a certificate.

b. Amy does not have a certificate, but has assisted Todd.

c. Rob has not assisted Sandra.

d. No one has assisted Joseph.

e. Everyone with a certificate has assisted another climber

Explanation / Answer

1. a. p = Bob is a statistician, q = Bob likes numbers, r = Bob is fun at parties

a. Bob likes numbers and he is a statistician.

q ^ p.

b. Although Bob likes numbers and he is a statistician, he is also fun at parties.

q ^ p ^ r

c. Bob likes numbers but he is not a statistician.

q ^ ~p

d. For Bob to be a statistician, it is necessary for him to like numbers and be boring at parties.

p -> (q ^ ~r)

e. Bob is a statistician if and only if he likes numbers and is fun at parties.

p <-> (q ^ r)

1.b. C(x) = x has a climbing certificate, A(x, y) = climber x has assisted climber y

a. Sylvia has a certificate.

C(Sylvia)

b. Amy does not have a certificate, but has assisted Todd.

~C(Amy) ^ A(Amy, Todd)

c. Rob has not assisted Sandra.

~A(Rob,Sandra)

d. No one has assisted Joseph.

~x (A(x,Joseph))

This is equivalent to

x (~A(x,Joseph))

e. Everyone with a certificate has assisted another climber

x(C(x) -> y (A(x,y)))

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