The University of Sydney
School of Mathematics and Statistics
Assignment 1
MATH1004: Discrete Mathematics Semester 2, 2022
Lecturer: Oded Yacobi
This individual assignment is due by 11:59pm Thursday 25 August 2022, via
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Copyright © 2022 The University of Sydney 1
1. Suppose you begin on a first rung of a ladder. A ladder path is a sequence of steps on
the ladder where each step is either up one rung or down one rung, you always remain
on the ladder, and you end back on the first rung. Assume the ladder is tall enough so
that you can never step off the top. For example, there is only one ladder path with two
steps, given by taking one step up and then one step down.
(a) Make a list of all the ladder paths with n steps, where n ∈ {3, 4, 5, 6}.
(b) Let n ≥ 1. Write down a correspondence that relates the ladder paths with 2n
steps to Catalan paths from (0, 0) to (n, n). More precisely, let Ln be the set of
ladder paths with 2n steps, and let Cn be the set of Catalan paths from (0, 0) to
(n, n). Construct a bijection f : Ln → Cn. (You don’t have to prove that it is a
bijection.)
2. (a) Find explicit bijections, and use the horizontal line tests to prove that they are
indeed bijections.
(i) f : (a, b) → (c, d), where a, b, c, d are real numbers such that a < b and
c < d.
(ii) g : (0, 1)→ (1,∞).
(b) For a set X, let
(
X
k
)
denote the set of subsets of X of cardinality k. Given n ≥ 1
we define three sets of cardinality n: Xn = {1, . . . , n}, X ′n = {1′, . . . , n′} and
X ′′n = {1′′, . . . , n′′}. For n > 3 construct a bijection
f :
(
Xn
n− 2
)
−→
(
Xn−1
n− 3
)
∪
(
X ′n−2
n− 3
)
∪
(
X ′′n−3
n− 3
)
,
and prove that it is a bijection.
2