Lecture Notes: Sets

Preliminaries

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Course Content

Tuesday Review

Question:

• Use strong induction to show that any amount of postage greater than or equal to 6¢ can be made using 2¢, 4¢, and 7¢ stamps.

Answer:

• Predicate: P(n) means “n¢ of postage can be made with 2¢, 4¢, and 7¢ stamps.
• Base case: Show P(n) is true for n = 6, 7.
• Induction hypothesis: Let \(n \geq 7\) be arbitrary but fixed, and assume \(P(m)\) holds whenever \( 6 \leq m \leq n\).
• Induction step: Given our I.H., We will prove \(P(n+1)\).
• By the I.H., \(P(n-1)\) is true, since \(n - 1 \geq n_0 = 6\). By adding a 2¢ stamp to the stamps used for \(P(n-1)\), we can measure \(n+1\) cents.

Thursday Review

Review Written Assignment 4.

Set Notation

• Write: Definition: A set is a collection of distinct objects, which are called its members.
• For example:
  \(\{1, 2, 4\}\)
  \(\{\{1, 2, 4\}\}\)
  \(\{\{1\}, \{2\}, \{4\}\}\)
• Ask: Are these sets equal?
• Nope! The first set has three members. The second set has one member. The third set has three members, each of which are sets.
• Sets don’t have order
• Sets don’t have repeated elements
• For example: Write the set of the class’s favorite ice cream flavors.

Special Sets

Subsets and Supersets

• Write: Definition: A subset of a set is a collection of elements drawn from the original set, but possibly missing some elements of the original.
• For example:
  \(\mathbb{N} \subseteq \mathbb{Z}\)
  \(\mathbb{Z} \subseteq \mathbb{Q}\)
  \(\emptyset \subseteq \mathbb{Z}\)
  \(\emptyset \subseteq \mathbb{Q}\)
  \(\mathbb{Z} \subseteq \mathbb{Z}\)
• Write: Definition: A proper subset is a subset which is not equal.
• For example:
  \(\mathbb{N} \subsetneq \mathbb{Z}\)
  \(\mathbb{Z} \subsetneq \mathbb{Q}\)
  \(\emptyset \subsetneq \mathbb{Z}\)
  \(\emptyset \subsetneq \mathbb{Q}\)
• Note that the notation \(\subset\) is ambiguous, so we will avoid it.
• Write: Definition: If A is a subset of B, then B is a superset of A.
• Write: Definition: If A is a proper subset of B, then B is a proper superset of A.

The Power Set

• Write: Definition: The power set of a set A is the set of all subsets of A.
• For example, if \(A = \{3, 17\}\) then \( \mathcal{P}(A) = \{ \emptyset, \{3\}, \{17\}, \{3, 17\}\} \)

Set Membership

• \( a \in S \) says that \( a\) is a member of the set \(S\).
• Ask: Are each of these true or false?
  \( 2 \in \{ 2, 5, 7\} \) is true.
  \( 3 \in \{ 2, 5, 7\} \) is false.
  \( 3 \not\in \{ 2, 5, 7\} \) is true.
  \( \mathbb{Z} \in \mathcal{P}(\mathbb{Q}) \) is true.
  \( 1 \subseteq \{ 1, 2\} \) is false.
  \( \{1\} \subseteq \{ 1, 2\} \) is true.
  \( \{1\} \subsetneq \{ 1, 2\} \) is true.
  \( \{1\} \in \{ 1, 2\} \) is false.
  

Set Cardinality

• Write: Definition: We denote the size of a set S, also called its cardinality, by \(\vert S \vert\).
• Suppose \(A = \{3, 17\}\)
• Ask: What is the cardinality of each of these sets?
  \( \vert \emptyset \vert = 0\)
  \( \vert A \vert = 2\)
  \( \vert \mathcal{P}(A) \vert = \vert \{ \emptyset, \{3\}, \{17\}, \{3, 17\}\} \vert = 4\)
  \( \vert \mathbb{Z} \vert = \infty\)
  \( \vert \{\mathbb{Z}\} \vert = 1\)

Set-builder Notation

• Write: The set of even integers can be written: \[ \{ n \in \mathbb{Z} : n \text{ is even} \} \] \[ \{ n \in \mathbb{Z} : n = 2m \text{ for some } m \in \mathbb{Z}\} \] \[ \{ 2m : m \in \mathbb{Z} \} \]
• This is called “set-builder notation.”
• The domain is written on the left, and the rule for inclusion in the set is written as a predicate on the right.
• More generally, we can form a subset of set A using a predicate P with the notation: \[ \{ x \in A : P(x) \} \]
• This subset will contain the elements of A which have property P.
• Note that in our third example, the domain was omitted from the left-hand side, and was specified in the rule on the right-hand side.

Set Operations

• Next, we will discuss operations you can perform on sets
• I will depict two sets with a Venn diagram
• Draw: Overlapping circles A and B, in two different colors
• First, I will color the intersection: \( A \cap B \)
• Next, the union: \( A \cup B \)
• Using set-builder notation, we can write union and intersection as: \[ A \cup B = \{x: x \in A \text{ or } x \in B \} \text{ (union)}\] \[ A \cap B = \{x: x \in A \text{ and } x \in B \} \text{ (intersect)}\]
• Union and intersection are associative, as are addition and multiplication of numbers. For sets A, B, C: \[ (A \cup B) \cup C = A \cup (B \cup C) \] \[ (A \cap B) \cap C = A \cap (B \cap C) \]
• This is just how “and” and “or” work in English
• For example: I have a cat who is “five years old and male, and orange”
  • It doesn’t matter where the comma goes
• Union and intersection are commutative. For sets A and B: \[ A \cup B = B \cup A \] \[ A \cap B = B \cap A \]
• We also have the set difference operation.
• Draw: \( A - B \) and \( B - A \).
• Note that: \( A \cup B = (A - B) \cup (B - A) \cup (A \cap B) \)
  • Point out each piece in the Venn diagram
• The complement of a set, \( \bar{B} = \{x : x \not\in B \} \). Draw it.
• Note that if the universe of \(x\) is not defined, it might be clear from context.
• But it is better to specify the universe, or else the reader might assume the universe is the universal set: \(U\)
• The universal set includes all possible elements!
• If you don’t specify the domain, then: \( \bar{B} = U - B \)
• For example: B = {Sat, Sun}. What is \( \bar{B} \)?
• Suppose that D = {Mon, Tue, Wed, Thur, Fri, Sat, Sun}.
• Then we could write: \[ B = \{x \in D : x \text{ is a weekend day} \} \] \[ \bar{B} = \{x \in D : x \text{ is a weekday} \} \] \[ \bar{B} = \{x \in D : x \not\in B \} \]

Disjoint

• Write: Definition: Sets A and B are disjoint if they have no elements in common.
• Ask: How to express this in mathematical notation?

\[ A \cap B = \emptyset \]

Distributive Laws

• This is the distributive law for addition and multiplication of numbers: \[ a \cdot (b+c) = a \cdot b + a \cdot c \]
• Similarly, there are distributive laws for union and intersection.
• Write: Distributive Laws \[ A \cup (B \cap C) = (A \cup B) \cap (A \cup C) \] \[ A \cap (B \cup C) = (A \cap B) \cup (A \cap C) \]
• Let’s take a closer look at the right-hand side of the first law
• Write: For an element \(x \in (A \cup B) \cap (A \cup C)\):
  • \( x \in (A \cup B) \) and
  • \( x \in (A \cup C) \)
• This means x is in A or (B and C), which is exactly what the left-hand side of the equation says.
• Let’s take a closer look at the left-hand side of the second law
• Write: For an element \(x \in A \cap (B \cup C)\):
  • \( x \in A \) and
  • \( x \in B \) or \( x \in C \)
• So x is in (A and B) or (A and C), which is exactly what the right-hand side of the equation says.

Skipping Limit Notation

This is on pages 53-54. We will introduce it when we need it.

Ordered Pairs

• In some cases, it is useful to have an ordered collection of elements.
• Write: For example, an ordered pair: \[ \langle x, y \rangle \] \[ \langle x, y \rangle = \langle z, w \rangle \text{ just in case } x = z \text{ and } y = w \]
• An ordered pair could represent points in a cartesian plane
• Draw: A plane with four points, with their coordinates labeled
• Write: A set containing those four points
• If you wanted to represent points in 3D space, you could use ordered triples: \( \langle x, y, z \rangle \)
• More generally, ordered n-tuples refers to a sequence of \( n \) elements.

Cross Product

• The last operation we will discuss today is called the Cartesian product or cross product.
• Write: Definition: For sets \( A, B \), the Cartesian product or cross product \( A \times B \) is the set of all ordered pairs with the first component from A and the second component from B.
• A cross product operates on sets, and yields a set of ordered pairs.
• For example, if A = {1, 2, 3} and B = {-1, -2} then: \[ A \times B = {⟨1, −1⟩, ⟨1, −2⟩, ⟨2, −1⟩, ⟨2, −2⟩, ⟨3, −1⟩, ⟨3, −2⟩} \]
• Ask: Does \( A \times B = B \times A \)?
• Answer: No, order matters.
• Ask: How to calculate \( \vert A \times B \vert \)?
• Answer: \( \vert A \times B \vert = \vert A \vert \cdot \vert B \vert \)
• The cross product is also defined for infinite sets.
• For example, I could denote regions of the cartesian plane using a cross product.
• Draw:
  • \( \mathbb{Z} \times \mathbb{Z} \), all integer coordinates.
  • \( \mathbb{N} \times \mathbb{N} \), all integer coordinates in the northeast.
  • \( \mathbb{Z} \times \mathbb{N} \), all integer coordinates in the north.
  • \( \mathbb{R} \times \mathbb{R} \), all points in the plane.