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Mathematics

Mathematics

1. Let R
+ = {x ∈ R : x > 0}. Consider the relation ρ in R
+ given by
xρy ⇐⇒
x
y
∈ Q
(a) Prove that ρ is an equivalence relation.
(b) Find [1] and [π]. Are two positive rational numbers equivalent under ρ? And two
positive irrational numbers?
2. New York City has a population of approximately 8,550,400 people. Prove that at least two
newyorkers share the same birthday.
3. Let Q = {n ∈ N : n = m
2
,m ∈ N}. Prove that |Q| = ℵ0.
4. Let σ be the permutation
σ =

1 2 3 4 5 6 7 8
4 2 7 5 1 3 8 6 !
.
(a) Find the decomposition of σ into cycles.
(b) Find the inverse σ
−1
and its decomposition into cycles.
(c) Find the parity of σ.
5. Let ∼ be the equivalence relation in Z given by n ∼ m ⇐⇒ n−m = 2k, k ∈ Z. We denote by
Z2 the quotient set Z∼.
(a) Let f : Z2 → Z and g : Z2 → Z2 be the correspondences given by f([n]) = 3n+1 and
g([n]) = [3n+1]. Are f and g well-defined?
(b) Let π : Z → Z2 be the projection into the quotient set given by π(x) = [x]. Find π(7)
and π
−1
([7]).
(c) Prove that the correspondence ◦ : Z2 ×Z2 → Z2 given by
[n] ◦ [m] = [n+m]
is well-defined and hence defines an operation in Z2.
(d) Is Z2 a group with the operation ◦?

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