Math220, Homework 2
Due in class January 18 or 19 (depending on your section)
1. Exercises for Chapter 1: Question 1.22. Let U = {1, 3, 5, . . . , 15} be the universal
set, let A = {1, 5, 9, 13}, and B = {3, 9, 15}. Determine the following:
(a) A ∪ B
(b) A ∩ B
(c) A − B
(d) B − A
(e) A
(f) A ∩ B.
Solution:
(a) A ∪ B = {1, 3, 5, 9, 13, 15}.
(b) A ∩ B = {9}.
(c) A − B = {1, 5, 13}.
(d) B − A = {3, 15}.
(e) A = {3, 7, 11, 15}.
(f) A ∩ B = {1, 5, 13} = A − B.
2. Exercises for Chapter 1: Question 1.26. (1 point each) Let U be the universal set
and let A, B be two subsets of U. Draw a Venn diagram for each of the following sets:
(a) A ∪ B
(b)
(a)
(b) A ∩ B
B
A
A
B
(c) A ∩ B
(d) A ∪ B
Solution:
(c)
The complement of A
is shaded blue, the
complement of B
shaded red, their
intersection is the
area that's both blue
and red.
(d)
A
A B
B
A
B
Here the union of the
complements of A and
B is the area that has
shading of any colour.
Note that these diagrams illustrate (but do not prove) DeMorgan laws: A ∪ B = A∩B
and A ∩ B = A ∪ B.
Page 1 of 3
Math220, Homework 2
Due in class January 18 or 19 (depending on your section)
3. Exercises for Chapter 1: Question 1.34. Give an example of two subsets A and B
of {1, 2, 3} such that all of the following are different: A ∪ B, A ∪ B, A ∪ B, A ∪ B, A ∩ B,
A ∩ B, A ∩ B, A ∩ B.
Solution: For example, take A={1}, B = {2}.
Then A ∪ B = {1, 2}, A ∪ B = {1, 3}, A ∪ B = {2, 3}, A ∪ B = {1, 2, 3}, A ∩ B = ∅,
A ∩ B = {1}, A ∩ B = {2}, A ∩ B = {3}.
4. Exercises for Chapter 1: Question 1.36. For aSreal number T
r, define Sr to be the
interval [r − 1, r + 2]. Let A = {1, 3, 4}. Determine α∈A Sα and α∈A Sα .
Solution: Plugging in r = 1, r = 3, and r = 4 into the definition of Sr , we get
S1 = [0, 3],
S3 = [2, 5],
(a)
S
α∈A
Sα = [0, 3] ∪ [2, 5] ∪ [3, 6] = [0, 6].
(b)
T
α∈A
Sα = [0, 3] ∩ [2, 5] ∩ [3, 6] = {3}.
S4 = [3, 6].
5. Let An = {n, n − 1} for every
natural number
n (so that An is a set of two elements for
S
T
each n ∈ N). Determine n∈N An and n∈N An .
S
Solution: We have n∈N An = {0, 1, 2, . . . } = N ∪ {0}. To prove it, first note that
every positive
integer n belongs to at least of of these sets, namely, to An , and 0 ∈ A1 .
S
Thus, n∈N An ⊇ {0, 1, 2, . . . } = N∪{0}. On the other hand, each set An is contained
in N ∪ {0}, and therefore their union is contained in this set as well.
T
Their intersection is empty: n∈N An = ∅, since the intersection of all these sets has
to be contained in the intersection of any two of them, and there exist two sets in
this collection whose intersection is empty: for example, if we take n = 1, we have
A1 = {0, 1}, and for n = 10, we have A10 = {9, 10}, so A1 ∩ A10 = ∅.
6. Exercises for Chapter 1: Question 1.64. For A = {1, 2} and B = {1}, determine
P(A × B).
Solution: By definition, A × B = {(1, 1), (2, 1)}.
Then P(A × B) = {∅, {(1, 1)}, {(2, 1)}, {(1, 1), (2, 1)}}.
7. Exercises for Chapter 1: Question 1.66.
For A = {a ∈ R : |a| ≤ 1} and B = {b ∈ R : |b| = 1}, give a geometric description of the
points in the xy-plane belonging to (A × B) ∪ (B × A).
Page 2 of 3
Math220, Homework 2
Due in class January 18 or 19 (depending on your section)
Solution: The set A × B, by definition, is the set of pairs A × B = {(a, 1), (a, −1) :
|a| ≤ 1}, so this is a union of two horizontal segments. On the other hand, B × A =
{(1, a), (−1, a) : |a| ≤ 1}, which is a union of two vertical segments. We get that
(A × B) ∪ (B × A) is a square with vertices (±1, ±1). (see picture).
Y
1
AxB is blue, BxA is red.
1
-1
X
-1
.
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