Math 220 Assignment 4
Due October 9th
Definition 1. LetSX and U be sets. For each x ∈ X, let Ax ⊆ U . Define the
X-indexed union
Ax = {u ∈ U : ∃x ∈ X 3 u ∈ Ax }. Similarly, define the
x∈X
T
X-indexed intersection
Ax = {u ∈ U : ∀x ∈ X, u ∈ Ax }.
x∈X
1. Let U be the set of positive integers between 1 and 50, inclusive. For an
integer d, let Ad be the multiples of d. Let X = {2, 3, 5}.
T
(a) What is
Ax ?
x∈X
(b) Let P be the set of prime numbers. What is U \ (P ∪
S
Ax )? Hint:
x∈X
it has exactly two elements.
2. Which of the following functions are equal?
(a) f : R \ {1} → R given by f (x) =
x3 −x
x−1 .
(b) g : R → R given by g(x) = x2 + x.
(c) h : R \ {1} → R given by h(x) = x2 + x.
(d) k : R → [−1, ∞) given by k(x) =
x3 −x
x−1 .
3. Let U = Z, and let Z≥2 denote the set of positive integers greater than or
equal to 2. For d ∈ Z+ , let Qd = {x ∈ Q : dx ∈ Z}.
(a) Prove that Q2 ∩ Q3 = Z.
S
(b) Prove that
Qd = Q.
d∈Z≥2
4. Let U and X be sets. Let A, B, C be subsets of U , and for each x ∈ X,
let Ax ⊆ U . Prove or disprove:
(a) U \ (A ∩ B ∩ C) = (U \ A) ∪ (U \ B) ∪ (U \ C).
T
S
(b) U \ (
Ax ) =
(U \ Ax ).
x∈X
x∈X
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5. Let O be the set of odd integers, and E be the set of even integers. Define
C : Z → Z by
n
n∈E
2
C(n) =
3n + 1 n ∈ O
(a) Prove that C(E) = Z.
(b) Prove that C(O) = {n ∈ Z : n%6 = 4}.
(c) Prove that C −1 (O) = {n ∈ Z : n%4 = 2}.
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