Website2025

Chapter 2Sets and Operations

Note 2.1: My long note

This is the NOTE environment!!

asd

Note 2.2: My long note

This is the NOTE environment!!

asd

Note 2.3: Test HMR refer to note-env

This is a test note to verify HMR works!

Note 2.4: Testing Hot Module Reload

This is a test note to verify HMR works!

Note 2.5: Test HMR refer to note-env-2

This is a test note to verify HMR works!

Note 2.6: Test HMR refer to note-env-2

This is a test note to verify HMR works!

Note 2.7: Test HMR refer to note-env-2

This is a test note to verify HMR works!

Note 2.8: Test HMR refer to note-env-2

This is a test note to verify HMR works!

Note 2.9: Test HMR refer to note-env-2

This is a test note to verify HMR works!

Note 2.10: Test HMR refer to note-env-2

This is a test note to verify HMR works!

Note 2.11: Test HMR refer to note-env-2

This is a test note to verify HMR works!

Note 2.12: Test HMR refer to note-env-2

This is a test note to verify HMR works!

Definition 2.1: My def

Some def

Definition 2.2: My def 2

Some def

Definition 2.3: My def 3

Some def

Here is the reference to the note environment: note-env[unresolved:note-env] and note-env-2[unresolved:note-env-2] and note-env-3[unresolved:note-env-3]

and [unresolved:note-env-4] and [unresolved:aaaq] and [unresolved:def2] and [unresolved:def3] and [unresolved:note-env-5] and [unresolved:note-env-6] and [unresolved:note-env-7] and [unresolved:note-env-8] and [unresolved:note-env-9] and [unresolved:note-env-a] and [unresolved:note-env-b], and [unresolved:note-env-c]

Section 2.1List examples

Unordered

Apple
Banana
Cherry

Numeric with custom separator and start

First item
Second item
Third item

Lettered and Roman

Alpha
Beta
Gamma

premise
step
conclusion
conclusion
conclusion
conclusion
conclusion
conclusion
conclusion
conclusion
conclusion
conclusion
conclusion
conclusion

Greek (fallback demo)

phase one
phase two
phase three

Deeply nested with defaults

Level 1 A
1. Level 2 a
  1. Level 3 i
    Level 4 A
    Level 4 B
    Level 5 i
    Level 5 ii
  2. Level 3 ii
2. Level 2 b
Level 1 B

Question 2.1: Basic membership

Let \(A = {1,2,3}\). Is \(2 \in A\)?

Question 2.2: Compare \(A \cup B\) and A ∩ B

Given sets \(A\) and \(B\), explain the difference between \(A \cup B\) and \(A \cap B\).

Question 2.3: Power set size

For a finite set \(A\) with \(|A| = n\), what is \(|(A)|\)?

Question 2.4: Determine whether A ⊆ B

Let \(A = {1,2}\) and \(B = {1,2,3}\). Is \(A \\subseteq B\)?

Question 2.5: Set identities with parts

Consider arbitrary sets \(A\), \(B\).

Difference vs intersection

State whether \((A \setminus B) \cap B = \varnothing\).

Answer

Symmetric difference

Express \(A \triangle B\) using unions and differences.

Solution

Idempotence

Prove \(A \cup A = A\).

Solution

Answer

Question 2.6: Pairs from small sets

Let \(A={1,2}\), \(B={x,y}\).

List A × B

Solution

List B × A

Answer

q-sample-6[unresolved:q-sample-6]

Todo 2.1: My long todo

This is the TODO environment!!

\sqrt[n]{xa}

and \(x\) and \(x^{22}\) Here is the reference to the todo environment: [unresolved:todo-env] , [unresolved:answer-env], and [unresolved:goal-env]

Goal 2.1: My goal

This is the GOAL environment!!

Goal 2.2: My answer

This is the ANSWER environment!!

\begin{bmatrix} 1 & 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4 \\ 1 & 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4 \\ 1 & 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4 \\ 1 & 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4 \\ \end{bmatrix}

Equation 2.2: The long matrix \(m\)

k = \begin{bmatrix} 1 & 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4 \\ 1 & 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4 \\ 1 & 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4 \\ 1 & 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4& 2& 3 & 4 \\ \end{bmatrix}

Equation 2.3: matrix \(k\)

  if (x < 44 && y > 2) { console.log(); }
  if (x < 44 && y > 2) { console.log(); }
  if (x < 44 && y > 2) { console.log(); }
  if (x < 44 && y > 2) { console.log(); }

4y-2z+6x=84y-2z+6x=84y-2z+6x=84y-2z+6x=84y-2z+6x=84y-2z+6x=84y-2z+6x=8 4y-2z+6x=84y-2z+6x=84y-2z+6x=84y-2z+6x=84y-2z+6x=84y-2z+6x=84y-2z+6x=84y-2z+6x=84y-2z+6x=84y-2z+6x=84y-2z+6x=84y-2z+6x=84y-2z+6x=84y-2z+6x=8

4y+7z+4x=34y+7z+4x=34y+7z+4x=34y+7z+4x=34y+7z+4x=34y+7z+4x=34y+7z+4x=3 4y-2z+6x=84y-2z+6x=84y-2z+6x=84y-2z+6x=84y-2z+6x=84y-2z+6x=84y-2z+6x=84y-2z+6x=84y-2z+6x=84y-2z+6x=84y-2z+6x=84y-2z+6x=84y-2z+6x=84y-2z+6x=8

3y-9z+1x=83y-9z+1x=83y-9z+1x=83y-9z+1x=83y-9z+1x=83y-9z+1x=83y-9z+1x=8 4y-2z+6x=84y-2z+6x=84y-2z+6x=84y-2z+6x=84y-2z+6x=84y-2z+6x=84y-2z+6x=84y-2z+6x=84y-2z+6x=84y-2z+6x=84y-2z+6x=84y-2z+6x=84y-2z+6x=84y-2z+6x=8

Equation 2.4: asdas A long title for the multiple equations to focus on.

The fist formula is: Equation 2.1, Equation 2.2 matrix \(m\), Equation 2.3 matrix \(k\). The main formula is Equation 2.4 Multiple equations to focus on., containing: Equation 2.4—i function r(){if(Ct(n),n.value===tl){let o=null;throw new T(-950,o)}return n.value}, Equation 2.4—ii function r(){if(Ct(n),n.value===tl){let o=null;throw new T(-950,o)}return n.value} and Equation 2.4—iii function r(){if(Ct(n),n.value===tl){let o=null;throw new T(-950,o)}return n.value}

Section 2.2Basic Concepts

Sets provide the language of modern mathematics. The operations introduced here power later topics like probability (see def-probability-measureSTAT1F92: Def Probability Measure) and the structure of the real line used by trigonometry (e.g., angle measure in def-radianCHEM1P00: Def Radian). Visual intuition can start with Venn diagrams such as fig-sets-vennDoc Figure 2.1 A Venn diagram for two sets A and B with overlapping region A ∩ B..

Definition 2.4: Empty Set \(x^2\)

The empty set, denoted ∅, is the unique set containing no elements.

Definition 2.5: Subset

Given sets A and B, we say A is a subset of B, written A ⊆ B, if every element of A is an element of B.

Definition 2.6: Union

The union of sets A and B is A ∪ B = x : x ∈ A or x ∈ B .

Definition 2.7: Intersection

The intersection of sets A and B is A ∩ B = x : x ∈ A and x ∈ B .

Definition 2.8: Set Difference

The difference A \ B = x : x ∈ A and x ∉ B contains the elements in A but not in B.

Definition 2.9: Power Set

The power set of A, denoted 𝒫(A), is the set of all subsets of A.

Two-set Venn diagram — **Figure 2.1.** A Venn diagram for two sets A and B with overlapping region A ∩ B.

Example 2.1: Basic Set Identities

For any sets A and B: (A ∪ B) \ B = A \ B and (A ∩ B) ⊆ A. These identities are visible in Doc Figure 2.1 A Venn diagram for two sets A and B with overlapping region A ∩ B..

Theorem 2.1: Inclusion–Exclusion for Two Sets

For finite sets A and B, one has |A ∪ B| = |A| + |B| − |A ∩ B|.

Proof 2.1:

Each element of A ∪ B is counted once in |A| + |B|, except elements in A ∩ B which are counted twice; subtracting |A ∩ B| corrects the overcount.

Section 2.2.1Symbol Summary

Table 2.1. Common set-theoretic symbols

Common set-theoretic symbols
Symbol	Name	Meaning
∅	Empty set	No elements
⊆	Subset	A ⊆ B means every element of A lies in B
∪	Union	Elements in A or B
∩	Intersection	Elements in both A and B
\	Difference	Elements in A but not in B
𝒫(A)	Power set	All subsets of A

Section 2.3Cartesian Products and Relations

Cartesian products support coordinates and functions—essential for geometry in ch-trigCHEM1P00: Ch Trig and for joint events in ch-probSTAT1F92: Ch Prob.

Definition 2.10: Cartesian Product

For sets A and B, the Cartesian product is A × B = (a, b) : a ∈ A, b ∈ B .

Definition 2.11: Relation

A relation on a set A is any subset R ⊆ A × A.

Definition 2.12: Equivalence Relation

A relation ~ on A is an equivalence relation if it is reflexive, symmetric, and transitive.

Example 2.2: Equivalence Classes

Congruence modulo n on ℤ is an equivalence relation; each class [a] contains integers differing by multiples of n.

Cartesian coordinate grid — **Figure 2.2.** A Cartesian grid for visualizing A × B as points in the plane.

Section 2.4Sigma-Algebras and Countability

The measurable structures used in probability (see def-probability-measure[unresolved:def-probability-measure]) are built from σ-algebras; countability arguments justify many constructions.

Definition 2.13: Sigma-Algebra

A σ-algebra 𝔽 on a set Ω is a nonempty family of subsets of Ω closed under complements and countable unions.

Theorem 2.2: Countable Union of Countable Sets

A countable union of countable sets is countable.

Proof 2.2:

Let Aₙ be countable sets. Enumerate each Aₙ as aₙk : k ≥ 1. Traverse the array (n, k) by diagonals and list each new element when first encountered; this yields an enumeration of ⋃ₙ Aₙ.

Example 2.3: Borel σ-algebra

The Borel sets on ℝ form the smallest σ-algebra containing all open intervals. They support integration, limits, and random variables (see [unresolved:def-random-variable]).

test test change