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Discrete Mathematics: An Open Introduction, 4th edition

Oscar Levin

Worksheet Preview Activity
A4US

In this preview activity, we will explore some basic properties of sets and functions. Later in this section, we will write proofs about these ideas.
1.
Remember that a set is just a collection of elements. Here are two definitions about sets:
  1. A set \(A\) is a subset of a set \(B\text{,}\) written \(A \subseteq B\) provided every element in \(A\) is also an element of \(B\text{.}\)
  2. Given sets \(A\) and \(B\text{,}\) the union of \(A\) and \(B\text{,}\) written \(A \cup B\) is the set containing every element that is in \(A\) or \(B\) or both.
Let’s build some examples.
(a)
Let \(B = \{1, 3, 5, 7, 9\}\text{.}\) Give an example of a set \(A\) containing \(3\) elements that is a subset of \(B\text{.}\)
What is \(A \cup B\) for the set \(A\) you gave as an example?
(b)
Give an example of two sets distinct sets \(A\) and \(B\) such that \(A \cup B = B\text{.}\)
\(A =\) ; \(B =\)
For the example you gave, is \(A \subseteq B\text{?}\)
  • yes
  • no
(c)
Find examples, if they exist, of sets \(A\) and \(B\) such that \(A \cup B \ne B\text{.}\)
\(A =\) ; \(B =\) .
For the example you gave, is \(A \subseteq B\text{?}\)
  • yes
  • no
2.
    Which of the following are always true?
  • For any sets \(A\) and \(B\text{,}\) \(A \cup B \subseteq B\text{.}\)
  • What if \(A = \{1,2,3\}\) and \(B = \{1, 3, 5\}\text{?}\)
  • For any sets \(A\) and \(B\text{,}\) \(B \subseteq A \cup B\text{.}\)
  • For any sets \(A\) and \(B\text{,}\) if \(A \subseteq B\text{,}\) then \(A \cup B \subseteq B\text{.}\)
  • For any sets \(A\) and \(B\text{,}\) if \(A \cup B = B\text{,}\) then \(A \subseteq B\text{.}\)
3.
For any function \(f: \N \to \N\) and any set \(A \subseteq \N\text{,}\) we can define the image of \(A\) under \(f\) to be the set of all outputs of \(f\) when the input is an element of \(A\text{.}\) We write this as \(f(A) = \{f(x) \st x \in A\}\text{.}\)
For the following tasks, let’s explore the function \(f: \N \to \N\) defined by \(f(x) = x^2 - 3x + 8\text{.}\)
(a)
Let \(A = \{1,2,3\}\) and \(B = \{2, 4, 6\}\text{.}\) Find \(f(A)\) and \(f(B)\text{.}\) Then find \(f(A) \cup f(B)\text{.}\)
\(f(A) =\) ; \(f(B) =\) ; \(f(A) \cup F(b) =\) .
(b)
Now find \(A \cup B\) and \(f(A \cup B)\text{.}\)
\(A \cup B =\) ; \(f(A \cup B) =\) .
(c)
Give an example, if one exists, of two distinct sets \(A\) and \(B\) such that \(A \subseteq B\) and \(f(A) \subseteq f(B)\text{.}\)
\(A =\); \(B =\).
Give an example, if one exists, of two distinct sets \(A\) and \(B\) such that \(A \subseteq B\) but \(f(A) \not\subseteq f(B)\text{.}\)
\(A =\); \(B =\).
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