Midterm — October 8th 2010
Mathematics 220
Page 1 of 6
This midterm has 4 questions on 6 pages, for a total of 25 points.
Duration: 50 minutes
• Read all the questions carefully before starting to work.
• With the exception of Q1, you should give complete arguments and explanations for
all your calculations; answers without justifications will not be marked.
• Continue on the back of the previous page if you run out of space.
• Attempt to answer all questions for partial credit.
• This is a closed-book examination. None of the following are allowed: documents,
cheat sheets or electronic devices of any kind (including calculators, cell phones, etc.)
Full Name (including all middle names):
Student-No:
Signature:
Question:
1
2
3
4
Total
Points:
7
6
6
6
25
Score:
Mathematics 220
7 marks
Midterm — October 8th 2010
Page 2 of 6
1. (a) Write the negation of the following statement:
“For every (a, b) ∈ N × N, a > b or a + b ≤ ab.”
(b) Write the negation of the following statement:
“If it is Friday then I have a midterm.”
(c) Write the converse and contrapositive of the following statement:
“If it is Tuesday then this is Belgium.”
(d) Give a precise mathematical definitions of the following sets
[
\
Sα
Sα
α∈I
(e) For any n ∈ N, let An =
B=
1
n
α∈I
, n2 + 1 . Simplify the following sets
[
n∈N
An
C=
\
An
n∈N
(f) Give an example of two sets, A and B, such that A ⊆ P (A) and B 6⊆ P (B).
Mathematics 220
6 marks
Midterm — October 8th 2010
Page 3 of 6
2. (a) Determine whether the following four statements are true or false — explain your
answers (“true” or “false” is not sufficient).
(i)
(ii)
(iii)
(iv)
∀x ∈ R, ∀y ∈ R, xy = x + y.
∀x ∈ R, ∃y ∈ R s.t. xy = x + y.
∃x ∈ R s.t. ∀y ∈ R, xy = x + y.
∃x ∈ R s.t. ∃y ∈ R s.t. xy = x + y.
(b) Prove or disprove the following statement
Let a, b, c, d ∈ R. If a ≤ c and b ≤ d then ab ≤ cd.
(c) Prove or disprove the following statement
For all n ∈ Z, there exists m ∈ Z such that
√
n2 = m
Mathematics 220
6 marks
Midterm — October 8th 2010
3. (a) Let n ∈ Z. Prove that n2 + 1 is even if and only if 7n + 3 is even.”
(b) Let S = {(x, y) ∈ R × R s.t. y = 3x − 7}. Prove the following
If (a, b) ∈ S and (b, a) ∈ S then a = b.
Page 4 of 6
Mathematics 220
6 marks
Midterm — October 8th 2010
4. Let A, B, C be sets.
(a) Prove that A ∪ B ⊆ A ∩ B.
(b) Prove that A ∩ (B ∪ C) ⊆ (A ∩ B) ∪ (A ∩ C).
Page 5 of 6
Mathematics 220
Midterm — October 8th 2010
This page has been left blank for your workings and solutions.
Page 6 of 6