MATH7861难点讨论

MATH7861 (S1, 2023) Assignment 1 Due: 4pm on 20 March 2023

All assignments in this course must be submitted electronically and SUBMITTED AS A SINGLE PDF FILE.

Prepare your assignment solutions using Word, LaTeX, Windows Journal, or other application, ensuring that

your name, student number and tutorial group number appear clearly at the top to the first page, and then

save your file in pdf format. Alternatively, you may handwrite your solutions and scan or photograph your

handwritten work to create a pdf file. Make sure that your pdf file is legible and that the file size is not

excessive. Use the assignment submission link in Blackboard to submit the pdf file.

1. (3 marks) Let p, q and r be statements. Construct a truth table for the following statement

form and determine whether it is a tautology, a contradiction, or neither of these.

(∼ (p ∨ (q →∼ r))) ↔ (r → (p∨ ∼ q))

2. (7 marks) Let p, q and r be statements.

(a) Use the Laws of Logical Equivalence and the equivalence of → to a disjunction to show

that:

(p → (q → r)) ∧ (∼ r) ≡ (p → ∼ q) ∧ (∼ r)

Show your working and name the laws that you use at each step.

(b) A MATH1061 student reads part (a) above and remarks,

“Take a close look at the two statements we need to show are logically equivalent.

Both statement include ∧ (∼ r) at the end. So it is probably fine to just show that

(p → (q → r)) is logically equivalent to (p → ∼ q).”

Do you agree with this student’s remark? Explain why or why not.

3. (5 marks) Determine whether the following argument is valid. Explain your reasoning.

If I go to work then I brush my teeth or drive my car.

I tie my shoes and I don’t wear socks.

If I wear socks then I don’t brush my teeth.

I go to work.

Therefore I drive my car or I don’t tie my shoes.

4. (4 marks) Rewrite each of the following sentences using quantifiers (∀, ∃) and mathematical

notation (∈, Z, ∧, →, ...) and state whether the sentence is true or false with a brief reason.

(a) A necessary condition for a real number to be rational is that its square is rational.

(b) A sufficient condition for a real number to be rational is that its square is rational.

5. (6 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) ∀q ∈ Q, ∃r ∈ Q such that q

r

∈ Z.

(b) ∃z ∈ Z such that ∀x, y ∈ Z, x − y < z.

Assignment continues next page.

MATH7861 (S1, 2023) Assignment 1 Due: 4pm on 20 March 2023

The following questions are based on material in Section 2.4 Application: Digital Logic Circuits

of the course textbook. See also the Tutorial Week 2 problem set and solutions.

6. (5 marks)

(a) Construct a Boolean expression having the given table as its truth table.

P Q R S output

(b) Simplify your Boolean expression from part (a) to obtain a logically equivalent Boolean

expression that uses exactly four logical connectives.

(c) Use your answer from part (b) to draw a digital logic circuit with exactly four logic gates

that has the table in part (a) as its input/output table.

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