MATH6005 Graduate Assignment A, 2023 ANU
Total Marks: 20 Value: 4% of final grade
Due: 2 pm Friday 10 March 2023
Please upload your solutions in PDF format, using the link provided. If you write the
solutions by hand, you will need to scan your work and save it as a pdf file.
Page 1 of your solutions document should be a ‘cover page’ containing only:
1. Title: “Graduate Assignment A”
2. Your full name.
3. Your ANU ID
4. The declaration: “I have read the ANU Academic Skills statement regarding collusion.”
(https://www.anu.edu.au/students/academic-skills/academic-integrity/plagiarism/collusion)
“I have not engaged in collusion in relation to this assignment”.
5. Your signature. (If you are typesetting rather than scanning a hand-written document, you
can type your name and it will be deemed a signature.)
6. The date and approximate time of your submission.
Regarding item 4, I emphasise the last paragraph of the Academic Skills statement:
The best way people can help each other to understand the material is to discuss the
ideas, questions, and potential solutions in general terms. However, students should
not draw up a detailed plan of their answers together. When it comes to
writing up the assignment, it should be done separately. If collusion is de-
tected, all students involved will receive no marks.
There are four questions. You may find some questions more difficult/time-consuming
than others, but nevertheless each question is worth the same (5 marks). The marking
criteria used to assess your work is be published on the course Wattle page in the same
place that the assignment is posted.
Page 1 of 5.
MATH6005 Graduate Assignment A, 2023 ANU
Question 1
(A) Draw a diagram of a circuit corresponding to the input-output table below.
X Y Z output
1 1 1 0
1 1 0 1
1 0 1 0
1 0 0 0
0 1 1 1
0 1 0 0
0 0 1 1
0 0 0 0
(B) Determine whether the following statement is true or false, and explain your rea-
soning:
The set {∧,∨} is functionally complete.
(C) A technical training manual in your workplace contains the following statement
Every compound statement is logically equivalent to one in which the
only symbols used are statement variables, parentheses, ‘↑’ and ‘?’.
A confused co-worker who is not enrolled in MATH6005 says “I have no idea what
that means, or why it may be helpful to know such a thing.” In no more than one
page, write an explanation for your coworker. Include some thoughts on why it is
a helpful thing to know.
Page 2 of 5.
MATH6005 Graduate Assignment A, 2023 ANU
Question 2
(A) Each of the variables in the following predicates is quantified over N:
p(x): x is prime d(t, x): t divides x
o(t): t = 1 q(t, x): t = x.
Using only quantifications, parentheses, logical connectives, variables and the pred-
icates d(t, x) and o(t) and q(t, x), write something in place of . . . in the following
to make a true statement. ?x ∈ N [p(x)? . . . ].
(B) Using only quantifications, parentheses, logical connectives, variables and the pred-
icates d(t, x) and o(t) and q(t, x) defined in part (A), write something in place of
. . . in the following to make a true statement.?x ∈ Z+ [?p(x)? . . . ].
(C) Let g ∶ Z→ Z be a function. Consider the following two statements, both assuming
the universe of integers.
Statement 1 ∶ x y([x ≤ y] ∧ [g(x) > g(y)])
Statement 2 ∶ y x([x ≤ y] ∧ [g(x) ≥ g(y)])
Without knowing any more about the function g, are you able to determine whether
or not Statement 1 is true? How about Statement 2? Explain your answers.
(D) Represent the form of the following argument using one of the notations introduced
in class, and then establish or refute the validity of the argument.
“There is no water on Planet X. If there is life on Planet X, then Planet X contains
water. Planet X will be invited to join the Confederation of Lively Planets only
if there is life on Planet X. Therefore, Planet X will not be invited to join the
Confederation of Lively Planets.”
(E) Sharky, a leader of the underworld, was killed by one of his own band of four min-
ions. Detective Sharp interviewed the minions and determined that all were lying
except for one. The detective’s notes from the interviews included the following:
? Socko said “Lefty killed Sharky.”
? Fats said“Muscles didn’t kill Sharky.”
? Lefty said “Muscles was shooting dice with Socko when Sharky was killed.”
? Muscles said “ Lefty didn’t kill Sharky.”
Who killed Sharky? Justify your answer.
Page 3 of 5.
MATH6005 Graduate Assignment A, 2023 ANU
Question 3
(A) Describe an example of each of the following: a set A containing 6 elements defined
using set-roster notation; a set B containing 27 elements defined using set-builder
notation; a partition of the integers that contains exactly four sets.
(B) Let A denote the lower-case English alphabet {a, b, c, . . . , z} and let X = A × Z ×
A × (Q ?Z). Use set-roster notation to give an example of a subset of X that has
exactly five elements.
(C) Use the logical equivalences below and the definitions of set operations to prove, or
provide a counterexample to disprove, the following set identity:
(A ∪B) ?C = (A ?C) ∪ (B ?C)
(D) Use the logical equivalences below and the definitions of set operations to prove, or
provide a counterexample to disprove, the following set identity:
(A ∪B) ?C = (A ?C) ∪B
(E) Write a paragraph to justify or refute the following opinion:
Sketching two Venn diagrams is helpful to decide whether or not a
set identity holds, but the Venn diagrams alone do not constitute a proof
that the identity holds.
The following is taken from the optional text. The text uses ～ instead of ? for
negation. You should use ?.
Page 4 of 5.
MATH6005 Graduate Assignment A, 2023 ANU
Question 4 A work colleague asks you the following in an email:
I have heard of Russell’s paradox, but I don’t know what it is. I think it
will come up in the client meeting tomorrow. Can you please explain Russell’s
paradox to me?
In no more than 500 words, write an email response to your colleague. An excellent
response will explain what Russell’s paradox is and how we remove the problem. An
excellent response will be written to be understood by someone who is quantitatively
curious and knows the very basics of set theory and its notation.
End of Questions for Assignment A
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