Numbers and Sets

Introductions

• Your preferred name and pronouns

• Why are you here?

  • Undergrads: why you signed up for this class
  • Grad students: why you’re doing the MA

• The last math class you took, and roughly when

• Your comfort food

Real-Number System

• Integers: \[...,-3,-2,-1,0,1,2,3,...\]

• Fractions: \[\frac{1}{2}, \frac{3}{5}, -\frac{2}{3}\]

• Rational numbers: ratio of integers

• Are fractions rational numbers? What about integers?

Real-Number System

• Rational numbers: ratio of integers
  • “terminating or repeating decimal”
  • Example. \(\frac{1}{3}=0.333\), \(\frac{1}{4}=0.25\)

Sets

• A set is a collection of distinct objects. \[A = \{brownies, icecream, pizza, ramen\}\]

• \(pizza \in A\), \(\in\) stands for ‘is in’

Set Relations

• Equivalence (\(=\)) \[A = \{brownies, icecream, pizza, ramen\}\] \[B = \{pizza, ramen, icecream, brownies\}\] \[A = B\]

Set Relations

• Disjoint sets \[A = \{brownies, icecream, pizza, ramen\}\] \[D = \{salad, fruits\}\]

Set Relations

• \(\emptyset\): empty or null set

Set Operations

• Union: \(A \cup B\), elements in either \(A\) or \(B\)

• Intersection: \(A \cap B\), elements in both \(A\) and \(B\)

• Example: \[A = \{brownies, icecream, pizza, ramen\}\] \[B = \{salad, fruits, icecream\}\]

  • \(A \cup B =\)
     
  • \(A \cap B =\)

Set Operations

• Union: \(A \cup B\), elements in either \(A\) or \(B\)

• Intersection: \(A \cap B\), elements in both \(A\) and \(B\)

• What about \[A = \{brownies, icecream, pizza, ramen\}\] \[B = \{salad, fruits\}\]

  • \(A \cup B =\)
     
  • \(A \cap B =\)

Set Operations

• Complement of \(A\): \(\tilde{A}\) or \(A^c\), ‘not \(A\)’

• Universal set \(U\) (context specific) then: \[A^c = \{x | x \in U \text{ and } x \not\in A\}\] Example. \(U=\{1,e,f,2\}, A=\{1,2\}\), then \(A^c=\{e,f\}\).

Set Operations: Venn Diagrams

Shade the following regions:

Laws of Set Operations

• Commutative law \[A \cup B = B \cup A \quad \quad A \cap B = B \cap A\]

• Distributive law \[A \cup (B \cap C) = (A \cup B) \cap (A \cup C)\] \[A \cap (B \cup C) = (A \cap B) \cup (A \cap C)\]

Distributive law

Verify by shading: \(A \cup (B \cap C) = (A \cup B) \cap (A \cup C)\)

Distributive law

Now the right-hand side, \((A \cup B) \cap (A \cup C)\):

Ordered Sets

• We said order does not matter for sets

• But we can have ordered sets where \[(a,b) \neq (b,a) \text{ unless } a=b\]

• Ordered pairs, triples,…

  • Example. \((age, weight)\), \((22,120)\) different from \((120,22)\)

Cartesian Product

The Cartesian product of two sets \(A\) and \(B\) is the set of all ordered pairs whose first element comes from \(A\) and second from \(B\): \[A \times B = \{(a,b) \mid a \in A, \; b \in B\}\]

Example. \(A = \{1,2\}\), \(B = \{3,4\}\) \[A \times B = \{ (1,3), (1,4), (2,3), (2,4) \}\]

Cartesian Plane

\[\mathbb{R}^2 = \{(x,y) | x \in \mathbb{R}, y \in \mathbb{R}\}\]

Can have \(\mathbb{R}^3, \mathbb{R}^4,...,\mathbb{R}^n\)

By the way

What is \(x \times x\)?

What is \(x^2 \times x\)?

What is \(x^2 \times x^2\)?