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MATH7861: Assignment
Assignment 1 must be submitted by 1 pm on Monday the 21st of March, 2022.
Submit your assignment using the TurnItIn link on the MATH1061/7861 Blackboard site.
For MATH7861 students this assignment is worth 10% of your final mark. Questions 1 to 4 will
be marked out of 30 and the essay question will be marked out of 15. Giving a total of 45. Note
that when the mark is entered into the blackboard site it will be scaled back to a mark out 30.
1. (5 marks) Construct a truth table for the following statement form, then state whether
or not the given statement form is a tautology.
[p→ (q → r)]←→ [(∼ p) ∨ ((∼ q) ∨ r)].
2. (5 marks) Write the negation of
(∀x ∈ Z)(∃y ∈ Z) such that x + y > 0.
Is the original statement true or false? Justify your answer.
3. (6 marks) Determine whether the following argument is valid.
If today is Saturday, then I do not ride my bike.
If I do not ride my bike, then I drive my car.
I drive my car and today is not Saturday.
Therefore I do ride my bike.
4. (14 marks) State whether or not the following statements are true or false. If the state-
ment is true prove it, if it is false give a counter example.
(i) ∀a, b ∈ R, √a + b = √a +√b.
(ii) ∀a, b ∈ Z, if 2a + b is even then a and b are even.
(iii) ∃a ∈ Z, s.t. (2a + 1)/2 is odd.
(iv) ∀n ∈ Z, if n2 is even then n is even.
4. (15 marks)
Short Essay ON
– Aristotelian diagrams and the square of opposition
– Euler circles
You are asked to choose from the above topics and write a 1.5 to 2 page essay that discusses
the related theory and applications of the topic you have chosen. If you wish you can
compare and contrast the two topics or just discuss one. Your discussion should include
historical aspects of the topic and how this topic relates to the mathematics studied in
MATH7861. Examples and diagrams of the relevant mathematics should be included.
Some references to get you started on the topic are available through the blackboard site.
Essay Structure
The essay should be a minimum of 1.5 pages and a maximum of 2.5 pages including a
bibliography. You can (but not required to) follow the following structure for you essay:
Background+History (1-2 paragraphs): In this section you can introduce the main
subject of this essay, discussing the historical aspects of the subject. (4 marks)
Analysis Section (3-4 paragraphs): What are the mathematical ideas associated with
this topic and how do they relate to the mathematics being studied in MATH7861?
Can you give examples of how this mathematics is relevant in our ever day lives? (5
marks)
Conclusion (1 paragraph): Conclude the essay by summarizing the ideas presented, if
possible discuss the impact of this mathematics on your current studies. (3 marks)
Bibliography: At least 3 sources (you can include the supplied article as a reference)
correctly cited in the essay and using a consistent citation format. (3 marks)