Assignment 5
SC/MATH 1190 A
(Fall 2023)
Part (a) Due Wednesday, October 25, at 1:30 pm
Part (b) Due Sunday, October 29, at 10:00 pm
Part (a): Pre-reading
Due Wednesday, October 25, at 1:30 pm
Answers for this assignment need to be submitted via Crowdmark. If provided with a textbox please
submit a short text answer: graphical answers, including pdfs, may not be graded when a textbox is
provided.
We will be finishing Chapter 2, and then skipping Chapter 3 in favour of moving on to Chapter 4.
• Section 2.5 (Cardinality of Sets). This section will tie together many of the themes from earlier
in the chapter. We will use the ideas of injections and surjections from Section 2.3 to extend the idea
of cardinality to infinite sets. We will say two sets have the same cardinality if there is a bijection
from one to the other. Inequality of cardinalities will be defined in terms of injections. This idea
agrees with all of our existing intuition about finite sets, but has the advantage that it also applies
to infinite sets, where some surprising results occus.
– Infinite sets with the same cardinality as N are said to be countable. This is the smallest
cardinality an infinite set can have. Many operations that make finite sets larger, turn out to
not affect the size of infinite sets. We will see that N, Z, N × N, and Q all have the same size.
– After we become convinced that there must only be one cardinality for infinite sets, we will see
that that intuition is also wrong. The set R is strictly larger than N, but the same size as both
[0, 1] and R × R. This has a profound implication to computer science, since, in particular, it
shows that there are real numbers it is impossible to describe with any computer program.
• Section 4.1 (Divisibility and Modular Arithmetic). In section 4.1, we will develop the ideas
of modular arithmetic (sometimes call clock arithmetic) that formalize a method for working with
remainders. Many of the results about even and odd numbers from Chapter 1 can be rephrased and
generalized in this setting. New notation that you might not be familiar with includes writing a|b to
mean that the integer b is divisible by a. We will also use the term mod in two different, but related
ways.
Question A1:
Evaluate 17 × 5 mod 80. Write your final answer in the provided text box.
Question A2:
We noted in the first week of class that it is possible to use a combination of 14¢ and 3¢ stamps to make
a total of exactly 90¢. One way to do this is with 3 × 14¢ plus 16 × 3¢. Which of the following statements
is true (select all that apply):
(a) 90 ≡ 16 × 3 (mod 14)
(b) 14 (90 − 16 × 3)
(c) 90 mod 14 = 3
(d) 90 ≡ 3 × 14 (mod 3)
(e) 90 and 3 × 14 have the same remainder when divided by 3
© 2023 Michael La Croix All Rights Reserved
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Part (b): Practice and Review
Due Sunday, October 29, at 10:00 pm
Please prepare long-form answers to each question in this part of the assignment.
Question B1:
Consider the sets A = {og, ot, awn, owl} and B = {d, l, c}.
(a) Use roster notation to list the elements of B × A.
Be careful about how you use parentheses and brackets. They mean different things.
(b) What is |B × A|? How does this relate to |A| and |B|?
Question B2:
Consider the set S = {a, b, c, d}.
(a) Write the elements of B = P(S).
Recall that P(S) is the power set of S.
(b) How many elements are in B?
(c) Every element of B is a finite set, and thus has a cardinality. Create a table showing how many
elements of B have each of the cardinalities 0, 1, 2, 3, and 4.
(d) We will now consider the polynomial (1 + a)(1 + b)(1 + c)(1 + d).
(i) Fully expand (1 + a)(1 + b)(1 + c)(1 + d).
(ii) How many terms are in your expanded polynomial? Describe how the terms are related to B.
(iii) If we let a = b = c = d = x then the polynomial takes the much simpler form (1 + x)4 . Fully
expand this polynomial, and collect like terms. How is this new polynomial related to your
table in (c)?
2
Note: To expand (1+x)4 , you could compute (1+x)2 . This is probably faster than expanding
by one factor at a time.
Question B3:
In class, we discussed a hypothetical situation where an airline uses a composition of functions to assign
a seat to passengers. If A is the set of all passengers, B is a set of flight itinerary numbers, and C is a
set of seats, the airline assigns flight itinerary numbers to passengers using some function f : A → B, and
then assigns seats to itinerary numbers using another function g : B → C. Their goal is to make sure that
there are never two passengers assigned to the same seat, which they can accomplish by making sure that
both f and g are injections, and assigning seats according to the function g ◦ f .
(a) Suppose the airline is not careful, and the function they use for f is not actually an injection.
(i) Unpack the definition of “not an injection” to say what this means (in terms of passengers and
flight itinerary numbers).
Hint: You should end up with a sentence that starts “There exist two passengers such that . . . ”
(ii) Show that under these circumstances, there must be two passengers assigned to the same seat.
(b) What if the airline was careful enough that f is an injection, but g is not? Is it possible that every
passenger is still assigned to a different seat from every other passenger?
© 2023 Michael La Croix All Rights Reserved
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