C Roettger, Spring ’12
Math 201 – Practice Exam 1 - solutions
Problem 1 Sketch the subsets of the plane
Solution A is a parallelogram with vertices (0, 0), (1, 2), (2, 1), (1, −1).
B is the upper semicircle of radius 2 about the origin.
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.
Solution a)
b) A ∪ (B ∩ C̄) (also possible 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.
Solution a) P ⇒ Q.
b) Whenever x is not a flibbertigibbet, every unicorn will love x.
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?
Solution Write E(n), O(n) for the statements that n is an even integer/odd
integer, respectively. a) ∼S = ∃x ∈ Z∀y ∈ Z : O(x2 + y 3 ) ∨ E(x + y).
b) There exists an integer x such that for all integers y, x2 + y 3 is odd or
x + y is even.
c) S is false. We prove this by distinguishing cases: Assume x is even. If y
is odd then y 3 is odd, x2 is even and hence x2 + y 3 is odd. If y is even, then
x + y is even. So no choice of y satisfies both parts of the ’and’-statement.
As a consequence, ∼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 statements in part b) by shading in a Venn diagram for each.
Solution a) and c) are below. For both, we have shaded the regions in the
diagram where the statement is not true.
b) Students who run at least 5 miles per week are not interested in Entomology or own at least one pet.
Alternative: If a student runs ..., then s/he is not interested in Entomology
or s/he owns at least one pet. Note the placement of the comma! it is meant
to show what parts are tied together by the ’or’.
R
R
P
E
P
E
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.
Solution Here are the four building-block statements. A: the agricultural
sector is shrinking.
B: zero-hedge bonds will rise above 4.7 percent yield.
C: the currency is devalued.
D: a discount rate of 0.6 percent is unsustainable (we’ll drop the filler ’completely’).
Now we can express the various clues using A, B, C, D:
1. C ⇒ A, 2. A ⇒ B, 3. B ⇒ D, 4. C∨ ∼A, 5. (A ∧ D) ∨ (C ∧ B).
In this form we could feed it to a computer program, failing that we use
paper and pencil to make a table. Either we use one line for each of the 16
possible combinations, or we use the 4x4-square as demonstrated in class.
I’ll demonstrate the latter. First, one diagram for each clue 1.-5. (row by
row, so clue 2. is shown top right). In every diagram, the shaded squares
represent combinations which are wrong (impossible) according to that clue.
In the last diagram, all shaded squares (impossible combinations) are put
together and leave only one possibility,
A ∧ B ∧ C ∧ D,
so all of these four statements are true.
About the labeling of the diagrams: A is the left half, ∼A the right half.
Similarly B is the top half, C the vertical middle and D the horizontal middle.
∼D is the top and bottom row, ∼C the leftmost and rightmost column.
A
A
B
B
D
D
C
C
A
A
B
B
D
D
C
C
A
A
B
B
D
C
D
C