SUNY Geneseo Department of Mathematics
Misc
Experimental survey for anonymously submitting questions to talk about in class and/or feedback on this course. See the “Questions and Suggestions” survey under “Quizzes” in Canvas.
Questions?
Subset and Equality Relationships?
How do the sets {1,2,3,4} and {{1,2,3},4} relate? How does either relate to {1,2,3}?
- {1,2,3} ⊆ {1,2,3,4} and {1,2,3} ⊂ {1,2,3,4} (the latter is the “proper subset” relation, i.e., subset but not equal)
- {1,2,3} ∈ {{1,2,3}, 4}
- { {1,2,3} } ⊆ {{1,2,3}, 4}
- {{1}} ∉ {{1,2,3}, 4}
Set Representation and Equality?
Order and number of occurrences don’t matter in sets
{1,2,3} = {3,2,1} = {1,2,3,2,2,1}
The Empty Set?
Empty truth sets show that something is false
Starting certain proofs from the empty set is really easy
“Null” instance of idea of set
Sets
(Parts of) Section 2.3
Roster Notation
Write the following sets in roster notation:
- The natural numbers greater than 1 but less than 6
- { 2, 3, 4, 5 }
- Pretty much the classical example of a set that can be listed in its entirety
- The primary colors
- { red, yellow, blue }
- Sets don’t have to be made from “mathematical” elements
- The perfect squares
- { 0, 1, 4, 9, 16, ... }
- An example of using “...” to show an infinite set in which some pattern defines the elements
- The integer multiples of π
- { ..., -3π, -2π, -π, 0, π, 2π, 3π, ... }
- A set that is infinite in two directions
Use roster notation to write a set of some characters from your favorite novel (or movie, etc.)
- { Legolas, Han Solo, Batman, Spiderman }
- { Sponge Bob, Patrick, Mr. Crabs, Gary }
- { Rick, Morty, Jerry, Summer }
- { Frodo, Legolas, Gandalf }
Equality and Subsets
Are any of the “favorite character” sets from above equal to each other? Are any subsets of another?
A set that is a subset of the character sets: { Batman, Spiderman }
How did you determine your answers to these questions?
- A ⊆ B: all elements of A are also elements of B
- i.e., to prove A ⊆ B, prove that if x ∈ A then x ∈ B
- A = B: all elements of A are also elements of B and vice versa
- To prove A = B, prove x ∈ A if and only if x ∈ B
- i.e., prove that if x ∈ A then x ∈ B, and then prove if x ∈ B then x ∈ A
Next
Predicates and sets
Read the rest of section 2.3 (i.e., “Variables and Open Sentences,” “Set Builder Notation,” and “When the Truth Set is the Universal Set” subsections)