MATH1061 Assignment 1 Due 1pm Thursday 26 March 2020
This Assignment is compulsory, and contributes 7.5% towards your final grade. It should be submitted by 1pm on Thursday 26 March, 2020.
Electronic submission Prepare your assignment as a single pdf file, either by typing it or by scanning 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.
1. (3 marks) Suppose that the statement form ((p∧ ∼ q)∨(p∧ ∼ r))∧(∼ p∨ ∼ s) is true. What can you conclude about the truth values of the variables p, q, r and s? Explain your reasoning.
2. (5 marks) Use the Laws of Logical Equivalence (provided in class and in the textbook page 35 of edition 4 and page 49 of edition 5) to show that:
((∼ (p ∨ ∼ q) ∨ (∼ p ∧ ∼ r)) ∧ s) ≡ ((r → q) ∧ ∼ (s → p)) where p, q, r and s are statements.
3. (6 marks) Consider the following argument:
Applying sunscreen, and wearing sun-smart clothing and a hat is sufficient for me to avoid sunburn. Avoiding sunburn is necessary in order for me to be comfortable.
I apply sunscreen and I am comfortable but I do not wear a hat.
Therefore I wear sun-smart clothing.
(a) Write this argument in symbolic form, using the variables listed below.
I apply sunscreen p I wear sun-smart clothing q I wear a hat r I avoid sunburn s I am comfortable u
(b) Determine whether the argument is valid or not. Show your working.
4. (4 marks) Use the rules of inference (provided in class and in the textbook page 61 of edition
4 and page XX of edition 5) to show that the following argument is valid.
∼s→r ∼ r →∼ p p∨∼s
q →∼ r
∴∼q
Assignment continues on the next page.
MATH1061 Assignment 1 Due 1pm Thursday 19 March 2020 5. (4 marks) Rewrite each of the following sentences using mathematical notation (∀, ∃, …) and
state whether the sentence is true or false with a brief reason. (a) If an integer is even, then it is divisible by 4.
(b) Every real number is greater than some integer.
6. (4 marks) Write down the negation of each of the following statements. Then determine whether
the statement or its negation is true, and explain why. (a) ∀x∈R,∃y∈R suchthatx2+y2 =16.
(b) ∀x,y∈R+,∃z∈Z+ suchthat x >z. y
7. (4 marks) Let P(x) and Q(x) be predicates and D be the domain of x. In each of the following situations, determine whether the two statements have the same truth value for every choice of P (x), Q(x) and D, or if there is a choice of P (x), Q(x) and D for which they have opposite truth values. Explain your reasoning.
(a) Statement 1: ∃x ∈ D such that (P (x) ∧ Q(x)).
Statement 2: (∃x ∈ D such that P (x)) ∧ (∃x ∈ D such that Q(x)).
(b) Statement 1: ∃x ∈ D such that (P (x) ∨ Q(x)).
Statement 2: (∃x ∈ D such that P (x)) ∨ (∃x ∈ D such that Q(x)).