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MATH1061

Assignment 4

This Assignment is compulsory, and contributes 5% towards your final grade. It should be submit-ted by 1pm on Thursday 28 October, 2021. In the absence of a medical certificate or other valid documented excuse, assignments submitted after the due date will incur a penalty as outlined in the course profile. Prepare your assignment as a pdf file, either by typing it, writing on a tablet or by scanning/photographing your handwritten work. Ensure that your name, student number and tutorial group number appear on the first page of your submission. Check that your pdf file is legible and that the file size is not excessive. Files that are poorly scanned and/or illegible may not be marked. Upload your submission using the assignment submission link in Blackboard. Please note that our online systems struggle with filenames that contain foreign characters (e.g. Chinese, Japanese, Arabic) so please ensure that your filename does not contain such characters.

1. (8 marks) Let S be the set of all functions having domain R and codomain R. Define a relation ρ on S as follows: ∀f, g ∈ S, (f, g) ∈ ρ if and only if f(x) − g(x) ∈ Z for all x ∈ R.

(a) Prove that the relation ρ is an equivalence relation.

(b) Consider the functions f, g ∈ S where f(x) = x and g(x) = x². Prove that [f]  [g].

(c) Let g ∈ S with g(x) = x². State two functions that are in the equivalence class [g].

2. (4 marks) Let τ be the partial order relation on the set S = {2, 4, 6, 8, 10, 12} defined as follows:

∀a, b ∈ S,    (a, b) ∈ τ if and only if a|b.

(a) Draw the Hasse diagram for the partial order τ .

(b) State a pair of noncomparable elements in this relation.

3. (6 marks) In Lecture 27, we showed that (Q − {1}, ⊗) is a group where the operation ⊗ is defined by a ⊗ b = a + b − ab for all a, b ∈ Q − {1}.

(a) Prove that {0, 2} is a subgroup of (Q − {1}, ⊗).

(b) Prove that {0, 2} is the only subgroup of (Q − {1}, ⊗) that has exactly two elements.

4. (5 marks) Recall that Sn is the group consisting of the set of all bijections from {1, 2, . . . , n} to {1, 2, . . . , n} together with the binary operation ○ denoting composition of functions. Determine the number of subgroups of S5 that are isomorphic to (Z3, +).

5. (9 marks) In Queensland, postal codes start with a 4 and consist of four digits, so they range from 4000 to 4999.

(a) How many Queensland postal codes are there?

(b) If you choose a Queensland postal code at random, what is the probability that it has four distinct digits?

(c) How many Queensland postal codes have only even digits or have exactly two occurrences of the digit 4? Explain your working.

(d) Determine the fewest number of Queensland postal codes that I must select to be guaranteed that I have at least two postal codes whose digits sum to the same number.