Mathematical Structures
Assignment 2
Tarik Mustafic
Number s1107262
Tutorial group x
September 13, 2023
Exercise B.1
a) f(n) = ±n (is / is not) a function from Z to R, because it does not define a unique input for an
integer.
b)
(is / is not) a function from Z to R, because when you solve the denominator for
zero you get 2 solutions
Exercise B.2
a) We consider the function that maps students in a discrete mathematics class to their mobile
phone number. This function is one-to-one if all of their phone numbers are unique to them
b) We consider the function that maps students in a discrete mathematics class to their student
identification number. This function is one-to-one if for every two students their indetification
number is unique
Exercise B.3
a) The function f : N → N is one-to-one but not onto, if we define it like
f(n) = 2n.
This is because it takes a natural number and multiplies it by two different inputs always
produce different outputs
b) The function f : N → N is both onto and one-to-one, if we define it like
f(n) = n+1
This is because it covers all the natural numbers so it is onto
Exercise B.4
• The function f(x) = ex from the set of real numbers to the set of real numbers is not invertible,
because it is not one-to-one
• The function f(x) = ex from the set of real numbers to the set of positive real numbers is
invertible, because ...
Exercise B.5
Let f be a function from A to B, where A and B are finite sets with |A| = |B |.
• Suppose that f is one-to-one, then f is onto, because •
Suppose that f is onto, then f is one-to-one, because ...
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Exercise B.6
a) The set of the integers greater than 10 is countably infinite, because this covers al integers
above 10
b) The set of the integers with absolute value less than 1,000,000 is infinite countably because
there are infinite integers smaller than 1,000,000
c) The set A × Z+ where A = {2,3} is infinite countably because there are an infinite number of
positive integers
d) The set of the integers that are multiples of 10 is infinite countable, because there are infinite
integers
Exercise B.7
a) We define the sets A and B:
def
A
= 2R
def
B
= R
Then A is uncountable because all real numbers are uncountable
And B is uncountable because all real numbers are uncountable
And A − B is countably infinite because 2 all real numbers – all real numbers are countable
b) We define the sets A and B:
def
A
=
R
def
B
=
all [0,1] real numbers
Then A is uncountable because all real numbers are uncountable
And B is uncountable because because all real numbers between 0 and 1 are uncountable
And A − B is uncountable because . all real numbers I larger than all real numbers between 0
and 1
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