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”
• Use this example for associativity of union and intersection
• 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.