Name:
Math 220 Problem Set 1 Answers
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1. Let A1 = 1, 2, 3, 4 , A2 = 1, 3, 5 , A3 = 2, 4 . For each of the following sets, use the “roster method”to specify its elements, i.e. explicitly list
the elements of the set between braces. The first one is done for you.
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(a) A1 ∩ A2 = 1, 3
(b) A1 ∩ A3 =
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2, 4
(c) A1 ∪ A3 =
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1, 2, 3, 4 = A1
(d) A1 − A3 =
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1, 3
(e) A3 − A1 = ∅
(f)
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Ai = A1 ∩ A2 ∩ A3 = ∅
i=1
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(g) P(A3 ) = {2, 4}, {2}, {4}, ∅
2. Let A1 =
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(x, y) ∈ R × R : x2 + y 2 = 1 be the unit circle centered at
the origin, ie. the setn of points in the plane whose
distance from (0, 0) is
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exactly 1. Let A2 = (x, y) ∈ R × R : y = 12 x be the line of slope 12 that
passes through the origin.
(a) Sketch A1 ∪ A2 .
(b) Use the “roster method”to specify the elements of A1 ∩ A2 .
Note that (x, x2 ) is an element of A1 ∩ A2 if and only if x2 + ( x2 )2 = 1.
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The two solutions to 5x4 = 1 are x = √25 and x = − √25 . hence
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A1 ∩ A2 = ( √25 , √15 ), (− √25 , − √15 )
3. For each positive integer n let An = [− n1 , n1 ] =
Find the real numbers a, b, c, d such that
10
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n=1
1
[c, d]. a = − 10
,b =
1
10 , c
n
x ∈ R : − n1 ≤ x ≤
An = [a, b] and
10
S
1
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.
An =
n=1
= −1, d = 1
Comment: Some people didn’t notice that An is a whole interval of real
numbers, not a finite set.
4. Let Z be the set of integers, and let Z be the set of real numbers that
are not integers. For each of the following sets S, give an example of an
element of S.
Comment: There are many correct answers for these. The most common
mistake was to give an alternative representation for the set S, instead of
answering the question by giving an example of some X such that X ∈ S.
(a) S = P(Z) = P owerset(Z)
X ∈ P(Z) if and only if X ⊆ Z. So any subset of Z would be
an acceptable answer. It could be finite or infinite. For example
X = {2, 3}. is one correct answer. Another correct answer is X =
the set of even integers = {n ∈ Z : n = 2d for some integer d} =
{0, 2, −2, 4, −4, 6, −6, . . . }.
(b) P(P(Z))
X ∈ P(P(Z)) iff and only if X ⊆ P(Z). Thus X can be any
(finiten or infinite) seto whose elements subsets of Z. For example
X = {1, 2, 3}, {1, 2} is an element of P(P(Z)).
(c) P(Z × Z)
The elements of Z×Z are ordered pairs of integers, hence the elements
of
are sets of ordered pairs of integers. For example X =
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(1, 2), (3, 4) ∈ P(Z × Z)
(d) Z × Z
( 12 , π) ∈ Z × Z.
(e) S = (R × R − Z × Z) − Z × Z.
A pair of real numbers is in S if and only if exactly one of the two
numbers is an integer. For example (2, π) ∈ S.
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