Assignment #1
COT-3100, Discrete Structures,
Fall 2021
Rules & Instructions:
• This assignment has 4 exercises (with multiple parts).
• Assignments must be turned in via Canvas.
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• Please name your submission as 1_XXXXXXX.zip, where XXXXXXX is your
seven-digit Panther ID number.
• Please submit all answers ONLY as Python and PDF files (using i.e. MS Word,
WordPerfect, TextMaker, LaTex, etc) for problem solutions. Photos and scanned
images, as well as solutions in a different format that the ones stated above
will be not accepted. From this rule, it is easy to infer that
o Handwritten answers will not be accepted due to legibility issues.
• Submissions turned in after the due date and/or which don’t meet the
established formatting rules will not be accepted.
Please remember that:
• I expect you to maintain a high level of academic integrity in this course and, if I
don't get it, there will be significant consequences.
• The basic principle to follow is that the work that you submit should be the result of
your own effort. It cannot be joint work with another student in the class. It cannot
e joint work with another student who previously took the class.
• You are allowed to consult other books in solving homework problems, or to discuss
the problems with other students, but you should cite any such sources that you use.
And you must write up your solutions on your own - merely copying answers from
other books or other students is definitely not acceptable.
Exercise #1: (35 pts = 7 pts x 5)
1.1. Find the pairs of equal sets over the real numbers, if any, give reasons:
A = {0}, B = {x | x > 15 and x < 5},
C = {x | x – 5 = 0}, D = {x | x2 = 25}
1.2. Let A, B, and C be three sets. If A B and B C, is it true that A C? If not, give an
example.
1.3. If A and B are two sets such that A B has 50 elements, A has 28 elements and B has
32, how many elements does A B have?
1.4. Show that A B = A B implies A = B.
1.5. If P(A) P(B), what is the relation between A and B?
Exercise #2: (30 pts = 6 pts x 5)
2.1. Define the relation R on the set of integer numbers (ℤ) by xRy if xy > 0. Is R an
equivalence relation? Justify.
2.2. Let R, the relation R on ℤ by aRb if a2 - b2 3. Is R an equivalence relation? Justify.
2.3. Let r be the relation on the set of integer numbers (ℤ) such that a
if and only if a + b
is odd. Is r an equivalence relation?
2.4. Let R be a relation defined on the set ℤ by aRb if a b. Is it true that R is symmetric and
transitive? If not, give an example.
2.5. Let A = {1, 2, 3, 4, 5}, and let
R = {(3,4), (5,5), (1,1), (2,2), (5,2), (1,4), (2,5), (3,1), (3,3), (4,1), (1,3), (4,3), (4,4)}
e an equivalence relation on A. Find the [1] and [3] on R.
Exercise #3: (20 pts = 5 pts x 4)
3.1. Consider the function f: R → R defined by f(x) = 2x +3. R – set of real numbers
(a) Find the inverse function of the function f.
(b) Is f ○ f-1 = f-1 ○ f?
3.2. Prove that the function f: R \ {2} → R \ {5} defined by f(x) =
5x+1
x−2
is bijective.
3.3. Determine if the function f: R → R defined by f(x) = x3 + 1 is a surjection (onto). R –
set of real numbers
3.4. Find the inverse function f: R \ {2} → R \ {1}, given by f(x) = (x + 1)/(x - 2)
Exercise #4: (15 pts = 3 pts + 5 pts + 7 pts)
A MySetOperations is a collection of functions, on which the following set operations are
defined:
myIsEmpty(A): Returns True if the set A is empty (A = ), False otherwise.
myIntersection(A,B): Returns the intersection of the set A and set B (A B).
myUnion(A,B): Returns the union of the set A and set B (A B).
Write a collection of MySetOperations in Python.
Note: To write these functions you must use only the flow and loop Python’s instructions, the
set build-in operations: in, not in, and add.