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Discrete Mathematics An Open Introduction, 4th Edition

Oscar Levin

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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.
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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 \(A \subseteq B\text{,}\) provided every element in \(A\) is also an element of \(B\text{.}\)
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  2. Given sets \(A\) and \(B\text{,}\) the union of \(A\) and \(B\text{,}\) written \(A \cup B\text{,}\) is the set containing every element that is in \(A\) or \(B\) or both.
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Let’s build some examples.
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(a)
Let \(B = \{1, 3, 5, 7, 9\}\text{.}\) Give an example of a set \(A\) containing \(3\) elements that is a subset of \(B\text{.}\)
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What is \(A \cup B\) for the set \(A\) you gave as an example?
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(b)
Give an example of two distinct sets \(A\) and \(B\) such that \(A \cup B = B\text{.}\)
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\(A =\) ; \(B =\)
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For the example you gave, is \(A \subseteq B\text{?}\)
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(c)
Find examples, if they exist, of sets \(A\) and \(B\) such that \(A \cup B \ne B\text{.}\)
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\(A =\) ; \(B =\) .
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For the example you gave, is \(A \subseteq B\text{?}\)
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2.
Which of the following are always true?
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  • For any sets \(A\) and \(B\text{,}\) \(A \cup B \subseteq B\text{.}\)
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  • What if \(A = \{1,2,3\}\) and \(B = \{1, 3, 5\}\text{?}\)
  • For any sets \(A\) and \(B\text{,}\) \(B \subseteq A \cup B\text{.}\)
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  • For any sets \(A\) and \(B\text{,}\) if \(A \subseteq B\text{,}\) then \(A \cup B \subseteq B\text{.}\)
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  • For any sets \(A\) and \(B\text{,}\) if \(A \cup B = B\text{,}\) then \(A \subseteq B\text{.}\)
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3.
For any function \(f: \mathbb{N} \to \mathbb{N}\) and any set \(A \subseteq \mathbb{N}\text{,}\) 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 \(f(A) = \{f(x) ~:~ x \in A\}\text{.}\)
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For the following tasks, let’s explore the function \(f: \mathbb{N} \to \mathbb{N}\) defined by \(f(x) = x^2 - 3x + 8\text{.}\)
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(a)
Let \(A = \{1,2,3\}\) and \(B = \{2, 4, 6\}\text{.}\) Find \(f(A)\) and \(f(B)\text{.}\) Then find \(f(A) \cup f(B)\text{.}\)
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\(f(A) =\) ; \(f(B) =\) ; \(f(A) \cup f(b) =\) .
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(b)
Now find \(A \cup B\) and \(f(A \cup B)\text{.}\)
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\(A \cup B =\) ; \(f(A \cup B) =\) .
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(c)
Give an example, if one exists, of two distinct sets \(A\) and \(B\) such that \(A \subseteq B\) and \(f(A) \subseteq f(B)\text{.}\)
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\(A =\); \(B =\).
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Give an example, if one exists, of two distinct sets \(A\) and \(B\) such that \(A \subseteq B\) but \(f(A) \not\subseteq f(B)\text{.}\)
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\(A =\); \(B =\).
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