C Roettger, Spring ’12
Math 201 – Practice Exam 1
Problem 1 Sketch the subsets of the plane
A = {(x, y) ∈ R2 : 0 ≤ x + y ≤ 3, 0 ≤ 2x − y ≤ 3},
[
√
B =
[−x, x] × [0, 4 − x2 ].
x∈[0,1]
Problem 2 a) Draw a truth table for the statement P ∧ (Q ∨ R).
b) Now assume that P, Q, R are all open sentences with a variable x. Let A,
B, C be the sets of all x such that P (x), resp. Q(x), R(x) are true. Express
the set of all x such that P (x) ∨ (Q(x)∧ ∼R(x)) is true using A, B, C.
Problem 3 a) Express the following statement in symbolic language: If
you think that the daffodils will start to come out next week, then I have
some seafront property in Boone to sell to you. Use the building blocks P
’You think the daffodils will start to come out next week’ and Q: I have some
seafront property in Boone to sell to you’.
b) Suppose R(x) is the statement: x is a flibbertigibbet, and Q(x) is: every
unicorn loves x. Express the statement ∀x : ((∼R(x)) ⇒ Q(x)) in English.
Problem 4 Consider the following statement S. For all x ∈ Z there exists y ∈ Z such that x2 + y 3 is an even number, and x + y is an odd number.
a) Express the negation ∼S of S using symbolic language, moving the negation as far inside as possible.
b) Express ∼S in plain English, but without any negative words.
c) Which of S, ∼S is true?
Problem 5 Consider the three subsets E, R, P of all ISU students who
are interested in Entomology (E), run at least 5 miles per week (R), and of
those who own at least one pet (P).
a) Illustrate the statement ’All students who are interested in Entomology,
but don’t own a pet, run at least 5 miles per week’ using a Venn diagram.
b) Express the statement R ⇒ (∼E ∨ P ) in plain English. Take care to make
the meaning as clear as possible.
c) Illustrate the above statements by shading in a Venn diagram for each.
Problem 6 Logicians don’t need to know about economics! but assuming these five clues, they can find out a lot about the economy.
If the currency is devalued, then the agricultural sector would be shrinking.
If the agricultural sector is shrinking, then zero-hedge bonds will rise above
4.7 percent yield.
If zero-hedge bonds rise above 4.7 percent yield, then a discount rate of 0.6
percent is completely unsustainable.
Maybe the currency will be devalued, or the agricultural sector will not
shrink.
Possibly the agricultural sector will shrink and a discount rate of 0.6 percent
is unsustainable, or the currency will be devalued and zero-hedge bonds will
rise above 4.7 percent yield.
Hints: a) find four ’building block’ statements and label them A, B, C, D so
that you can write each clue in symbolic form, using only these four letters.
b) You can ignore the difference between ’is’ and ’will be/would’, present and
future. c) The last comma is very important.