Chapter 1: Set Theory - Lecture Notes

A set of practice Q&A flashcards covering core concepts from the lecture notes on Course Logistics and Chapter 1: Set Theory.

What is the primary purpose of Set Theory in mathematics as described in the notes?

Foundational and provides concrete definitions in mathematics.

How is a set denoted and how is an element relation written?

Sets are denoted by capital letters (e.g., A); element relation written as x ∈ A, and x ∉ A means not in.

What does x ∈ A mean?

Element x is a member of set A.

What does x ∉ A mean?

Element x is not a member of set A.

What is enumeration in describing a set?

Listing all elements inside curly braces, e.g., {0, 1} for the set of binary digits.

Give an example of an enumerated set for decimal digits.

{0,1,2,3,4,5,6,7,8,9}

How are infinite or large sets often shown succinctly in the notes?

Using ellipses '…' to indicate continuation, e.g., {1, 2, …, 100} or {1, 2, …} for positive integers.

What do the symbols N, Z, Q, R, and C represent?

N = natural numbers; Z = integers; Q = rational numbers; R = real numbers; C = complex numbers; with N ⊆ Z ⊆ Q ⊆ R ⊆ C.

What is Set Builder Notation?

A way to define sets by a rule, e.g., {a/b | a ∈ Z and b ∈ Z and b ≠ 0}.

What does the symbol '|' mean in Set Builder Notation?

It means 'such that' (or ':' in some texts).

What is a Finite Set and what is Cardinality?

Finite Set: a set with a finite number of elements. |A| denotes the number of distinct elements in A.

What is a Subset (A ⊆ B)?

A is a subset of B if every element of A is also an element of B.

How are sets ordered in the typical inclusion chain of number systems?

N ⊆ Z ⊆ Q ⊆ R ⊆ C.

What is Set Equality?

A = B iff every element of A is in B and every element of B is in A (equivalently A ⊆ B and B ⊆ A).

Do multiplicity and order matter in a set?

No. Sets ignore repetition and order; e.g., {5,5,5,1} = {5,1} and {1,5}.

What is the Empty Set?

The set with no elements, denoted ∅ or {}; it is a subset of every set.

What is the difference between improper and proper subsets?

A ⊆ A is an improper subset. A is a proper subset of B if A ⊆ B and B has at least one element not in A.

What are some key takeaways for mastering set theory from the notes?

Understand core definitions and notation; practice converting between enumeration and set-builder notation; remember the empty set is a subset of every set; use homework solutions; and use LaTeX/Overleaf for notation.