辅导案例-MAST30012-Assignment 2

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School of Mathematics and Statistics
MAST30012 Discrete Mathematics,
Assignment 2,
Semester 2, 2020
Student Name Student Number
Tutor’s Name Practice Class Day/Time
Submission deadline is 23:59pm Monday 21 September.
Check Canvas for details of the late submission policy.
Submit your assignment online using Canvas as a single pdf
including this cover page (more details to follow).
• Full working must be shown in your solutions.
• Marks will be deducted for incomplete working, insufficient justification or incorrect notation.
Q1: Give a bijective proof of the following identity!
k≥0
"
n
2k
#
= 2n−1
using the following steps:
(a) Use subsets of the set [n] to define a set ΩL representing the left-hand side.
(b) Define a set ΩR representing the right-hand side as a certain set of tuples with entries from the
set {0, 1}.
(c) Define a bijective function Γ : ΩR → ΩL. (Note the direction of the function must be right-to-
left).
(d) Show that your function in (c) is well defined. (There is no need to prove it is a bijection).
Your definitions in (a) and (b) must use the specification form of a set definition (see week 1 lecture
notes).
Q2: Select n2 + 1 distinct points in (or on the boundary of) the unit square. Prove that at least two
points are no more than a distance

2/n apart.
End of Assignment.

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