202201 Math 122 Assignment 2

Due: Friday, Feb. 11, 2022 at 23:59. Please submit on your section’s Crowdmark page.

There are five questions of equal value (worth a total of 45 marks), and one bonus question (worth

4 marks). Please feel free to discuss these problems with each other. You may not access any

“tutoring” or “help” website in any way. In the end, each person must write up their own solution,

in their own words, in a way that reflects their own understanding. Complete solutions are those

which are coherently written, and include appropriate justifications.

1. First write the argument below in symbolic form, and then determine if it is valid or invalid.

If the argument is valid, prove it using known logical equivalences and inference rules. If it is

invalid, demonstrate that by giving a counterexample.

If I go motorcycling, then it rains. If it rains and is warm, then I wear rain gear or decide

to get wet. If my friends phone me to go riding, then I go motorcycling and it is warm. I do

not decide to get wet. Therefore, if my friends phone me to go riding, then I wear rain gear.

2. (a) Suppose the universe is the real numbers. Write the following statement in English, and

also determine its truth value. Do not literally translate the symbols, as in “For all x,

there exists y, such that ...” Instead, write an understandable sentence in plain English

that accurately describes the mathematical property being precisely specified, and that

starts with “Every positive real number has ...”.

∀x, (x > 0)→ [(∃y1, ∃y2, (y21 = x) ∧ (y22 = x) ∧ (y2 6= y1))

∧(∀z, (z 6= y1) ∧ (z 6= y2)→ (z2 6= x))].

(b) Write the sentence “there is no smallest integer” in symbols, making all quantifiers

explicit. Remember to state the universe.

(c) Suppose the universe is the integers. Write the negation of ∀n, ∃m,m ·n = n in symbols,

with quantifiers, and without using negation (¬) or any negated mathematical symbols

like 6= or 6>.

3. Consider the two statements s1 : ∀n, p(n) ∨ q(n) and s2 : [∀n, p(n)] ∨ [∀n, q(n)].

(a) Explain why s2 logically implies s1.

(b) Are s1 and s2 logically equivalent (no matter what universe and the open statements

p(n) and q(n))? Why of why not?

(c) Are the statements s1 : ∃n, p(n)∨q(n) and s2 : [∃n, p(n)]∨ [∃n, q(n)] logically equivalent?

Why pr why not?

4. (a) Let n be an integer. Prove that if n4 is a multiple of 3 then n is a multiple of 3 by stating

the contrapositive, and then proving it by cases. (The possible remainders on division

y 3 are 0, 1 and 2.)

(b) Prove that

√

8 is i

ational. (Hint: look at the co

esponding proof fo

√

2.)

(c) Use the fact that

√

2 is i

ational to show that

√

22m+1 is i

ational for any positive

integer m.

5. Answer each question True or False, and write a sentence or two to

iefly explain you

easoning. Let A = {1, {1}, 2, {∅}, {{1}, {2}}, {{1}, 2}}.

(a) {2} ∈ A

(b) {1, 2} $ A

(c) {{1, {2}}} ⊆ A

(d) ∅ ∈ A

(e) A ∩ P(A) = ∅

(f) {2} ∈ P(A).

(g) ∅ is the only set with no non-empty proper subset.

6. (Bonus question, 4 marks) Suppose you are given a 2 × 2 a

ay of lights such that for each

ow of the a

ay there is a switch that changes the state of every light in the row (from off

to on, and on to off), and similarly for each column. Explain why the following statement is

true: The switches can be flipped so that all lights are eventually on if and only if there are

an even number of lights on. Suppose you are given a similar 3 × 3 a

ay of lights with two

lights on in each row. Is it always possible to flip the switches so that all lights are eventually

off? Why or why not?

Answered 3 days AfterFeb 08, 2022

CamScanner 02-11-2022 05.26.31

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