Equivalence Relations - Reflexive, Symmetric, and Transitive

1. Caption evidence
   Observation

   On-screen text box reads: "An equivalence relation on a set is a relation that is" followed by bullets "Reflexive", "Symmetric", "Transitive".

2. Audio
   Observation

   The lecturer says that when a relation has all three properties, "we're going to call it an equivalence relation."

equivalence relation

A relation on a set that is reflexive, symmetric, and transitive.

Relations on sets

1. Caption evidence
   Observation

   Bullet 1 in the definition box: "Reflexive".

2. Audio
   Observation

   Lecturer lists "they are reflexive" as one of the three required properties.

Reflexive

Property of a relation requiring every element to be related to itself; illustrated verbally with $3 = 3$.

Binary relations

1. Caption evidence
   Observation

   Bullet 2 in the definition box: "Symmetric".

2. Audio
   Observation

   Lecturer lists "they are symmetric" and later demonstrates reordering $3 = 3$.

Symmetric

Property of a relation allowing the two sides of a related pair to be swapped while preserving the relation.

Binary relations

1. Caption evidence
   Observation

   Bullet 3 in the definition box: "Transitive".

2. Audio
   Observation

   Lecturer lists "they are transitive" and later chains equalities involving 3.

Transitive

Property of a relation where if one element relates to a second and the second relates to a third, then the first relates to the third.

Binary relations

1. Formula
   Observation

   Handwritten expression appears as "10 mod 3".

2. Audio
   Observation

   Lecturer says "how about 10 mod 3" and explains it as the remainder when 10 is divided by 3.

mod

Modulo operator giving the remainder after division.

Integers (as used in the example)

1. Formula
   Observation

   Written as the left operand in "10 mod 3".

2. Audio
   Observation

   Spoken as part of the example "10 mod 3".

10

Dividend in the modular arithmetic example.

Integer constant

1. Formula
   Observation

   Written as the right operand in "10 mod 3".

2. Audio
   Observation

   Spoken as the divisor in "10 mod 3".

3

Divisor/modulus in the modular arithmetic example.

Positive integer constant

1. Audio
   Observation

   Lecturer states "10 mod 3 is going to be the value of 1" and calls it the remainder.

1

Remainder of 10 upon division by 3.

Integer result

1. Formula
   Observation

   The notation 'a mod b' is written on the board multiple times, such as '10 mod 3'.

$a \bmod b$

Modulo operation, representing the remainder when a is divided by b.

Integers

1. Formula
   Observation

   The expression 'mod 3' appears repeatedly in the handwritten examples on the board.

$\text{mod } 3$

Modulo 3 operation, which finds the remainder when a number is divided by 3.

Integers

1. Caption evidence
   Observation

   Definition box explicitly names the concept and lists three bullet properties.

2. Audio
   Observation

   Lecturer introduces the term and ties it to having all three properties.

An equivalence relation is a relation on a set that simultaneously satisfies three properties: reflexivity, symmetry, and transitivity. The video presents this as the central named concept of the segment.

1. Defined for a relation on a set

2. Requires all three listed properties at once

1. Audio
   Observation

   Lecturer says equality is "a way ... of relating numbers on the left with numbers on the right" and calls it a relation on numbers.

The lecturer treats ordinary equality between numbers as a relation: it connects a number on the left with a number on the right. This reframing is used to show that equality fits the abstract notion of a relation before checking the three equivalence properties.

1. Applies to numbers (example uses integers)

1. Definition of equivalence relation

1. Audio
   Observation

   Lecturer announces a second example, "equality of modular arithmetic," then writes and discusses "10 mod 3".

2. Formula
   Observation

   Handwritten "10 mod 3" appears on screen.

1. The full verification that modular equality is an equivalence relation is not completed within this 120-second clip.

After using ordinary equality as the first example, the lecturer introduces modular arithmetic as a second example of an equivalence relation. The method shown is to start from a concrete modulo expression, compute its remainder, and use that as the basis for discussing equality modulo a number.

1. Uses a concrete integer example

2. Clip ends before the broader equivalence-relation verification is fully developed

1. Definition of equivalence relation

1. Diagram
   Observation

   A text box on the screen defines an equivalence relation.

An equivalence relation on a set is a relation that satisfies three properties: it is reflexive, symmetric, and transitive.

1. Must be defined on a set

2. Must satisfy reflexivity

3. Must satisfy symmetry

4. Must satisfy transitivity

1. Audio
   Observation

   The speaker explains the reflexive point using the example 10 mod $3 = 10$ mod 3.

2. Formula
   Observation

   Written on the board: 'R: 10 mod $3 = 10$ mod 3'.

A relation is reflexive if every element is related to itself. In modular arithmetic, this means a mod $n = a$ mod n for any integer a.

1. n is a positive integer

2. a is an integer

1. Definition of Equivalence Relation

1. Audio
   Observation

   The speaker demonstrates symmetry by showing that if 10 mod $3 = 7$ mod 3, then 7 mod $3 = 10$ mod 3.

2. Formula
   Observation

   Written on the board: 'S: 10 mod $3 = 7$ mod 3 => 7 mod $3 = 10$ mod 3'.

A relation is symmetric if whenever x is related to y, y is also related to x. In modular arithmetic, if a mod $n = b$ mod n, then b mod $n = a$ mod n.

1. n is a positive integer

2. a and b are integers

1. Definition of Equivalence Relation

1. Audio
   Observation

   The speaker begins to explain transitivity by introducing a third number, 13, which also has a remainder of 1 when divided by 3.

2. Formula
   Observation

   Written on the board: 'T: 13 mod 3'.

1. The full derivation for transitivity is not completed within the provided video segment.

A relation is transitive if whenever x is related to y and y is related to z, then x is related to z. The speaker sets up an example with 10, 7, and 13 to demonstrate this.

1. n is a positive integer

2. a, b, and c are integers

1. Definition of Equivalence Relation

1. Caption evidence
   Observation

   An equivalence relation on a set is a relation that is Reflexive, Symmetric, Transitive.

An equivalence relation on a set is defined as a relation that satisfies three properties: it must be reflexive, symmetric, and transitive.

1. The relation must be reflexive.

2. The relation must be symmetric.

3. The relation must be transitive.

1. Audio
   Observation

   Well, if you have that 10 mod 3 equals 7 mod 3, and you have that 7 mod 3 equals 13 mod 3... then the 10 mod 3 equals the 13 mod 3 as well. And so you can get transitivity for your modular arithmetic.

2. Formula
   Observation

   Handwritten text shows 'T: 13 mod 3' with arrows pointing to previous equations.

Transitivity is demonstrated using specific numbers: if 10 mod 3 equals 7 mod 3, and 7 mod 3 equals 13 mod 3, then it follows that 10 mod 3 equals 13 mod 3.

1. The relation is equality of remainders modulo 3.

1. Definition of an Equivalence Relation

1. Caption evidence
   Observation

   Text box states: "An equivalence relation on a set is a relation that is" with bullets "Reflexive", "Symmetric", "Transitive".

2. Audio
   Observation

   Lecturer says that when we have all three properties, we call it an equivalence relation.

A relation on a set is an equivalence relation exactly when it is reflexive, symmetric, and transitive.

1. The object under discussion is a relation on a set.

For a given relation on a set, the three properties must all hold.

1. Audio
   Observation

   Lecturer checks reflexivity, symmetry, and transitivity for equality using examples with 3, then concludes: "So equality is our notion of an equivalence relation."

Ordinary equality on numbers is an equivalence relation because it is reflexive, symmetric, and transitive.

1. Equality is viewed as a relation on numbers.

Illustrated with the specific example 3, but stated generally for equality.

1. Audio
   Observation

   Lecturer says "10 mod 3 is going to be the value of 1" and identifies it as the remainder when 10 is divided by 3.

2. Formula
   Observation

   The written setup is "10 mod 3".

10 mod 3 equals 1.

1. mod is interpreted as the remainder operation from division.

Specific numerical claim.

1. Audio
   Observation

   The lecturer verbally walks through each property in order using the number 3 as the example.

1. The argument is presented informally with a single numeric example rather than a fully symbolic general proof.

1. Expression

   $$3 = 3$$
   Explanation

   The lecturer begins with the concrete statement that three is always equal to three.

   Justification

   Used to illustrate reflexivity of equality.

   Shown in the video
2. Expression

   $$3 = 3 \text{ reordered is still } 3 = 3$$
   Explanation

   He then says that if the two sides are reordered, the same equality still holds.

   Justification

   Used to illustrate symmetry of equality.

   Shown in the video
3. Expression

   $$3 = 3 \text{ and } 3 = 3 \Rightarrow 3 = 3$$
   Explanation

   Finally he chains two equalities and concludes that the original 3 equals the third 3.

   Justification

   Used to illustrate transitivity of equality.

   Shown in the video

Equality is an equivalence relation because it satisfies reflexivity, symmetry, and transitivity.

1. Formula
   Observation

   Handwritten expression "10 mod 3" is shown.

2. Audio
   Observation

   Lecturer explains the value as the remainder when 10 is divided by 3.

1. The final spoken sentence is cut off mid-explanation after "you can say 10 is equal to 3...".

1. Expression

   $$10 \bmod 3$$
   Explanation

   The lecturer introduces a concrete modular arithmetic expression.

   Justification

   Setup for the second example.

   Shown in the video
2. Expression

   $$10 \bmod 3 = 1$$
   Explanation

   He states that the value is 1.

   Justification

   Interpreted as the remainder after dividing 10 by 3.

   Shown in the video
3. Expression

   $$10 = 3 \cdot 3 + 1$$
   Explanation

   He begins to justify the remainder by rewriting 10 in terms of division by 3, but the sentence is incomplete in the clip.

   Justification

   Standard division-with-remainder explanation; only partially audible here.

   Shown in the video

In the example, 10 mod 3 equals 1 because 1 is the remainder when 10 is divided by 3.

1. Audio
   Observation

   Speaker states 'Doing the reflexive point is pretty trivial here... we're just going to be saying that this is exactly the same thing as 10 modulo 3, which is also equal to 1.'

2. Formula
   Observation

   Board shows '10 mod $3 = 10$ mod 3' with '1' written below both sides.

1. Expression

   $$10 = 3 \times 3 + 1$$
   Explanation

   Express 10 in terms of division by 3.

   Justification

   Definition of modulo operation.

   Derived from the video
2. Expression

   $$10 \bmod 3 = 1$$
   Explanation

   Calculate the remainder of 10 divided by 3.

   Justification

   From the previous step.

   Shown in the video
3. Expression

   $$10 \bmod 3 = 10 \bmod 3$$
   Explanation

   State the reflexive property for the specific example.

   Justification

   Any value is equal to itself.

   Shown in the video

The relation is reflexive because 10 mod 3 is equal to itself.

1. Audio
   Observation

   Speaker says 'If I look at 10 modulo 3... how about 7 modulo 3? Indeed, 7 is twice 3 plus 1... So all of these turn out to be equal to these values just of 1.' Then, 'if I flip this around and try to write 7 mod 3, well, that is also equal to 10 mod 3.'

2. Formula
   Observation

   Board shows 'S: 10 mod $3 = 7$ mod 3 => 7 mod $3 = 10$ mod 3' with '1' written below each term.

1. Expression

   $$10 \bmod 3 = 1$$
   Explanation

   Establish the value of the first term.

   Justification

   Previously calculated.

   Shown in the video
2. Expression

   $$7 = 2 \times 3 + 1$$
   Explanation

   Express 7 in terms of division by 3.

   Justification

   Definition of modulo operation.

   Shown in the video
3. Expression

   $$7 \bmod 3 = 1$$
   Explanation

   Calculate the remainder of 7 divided by 3.

   Justification

   From the previous step.

   Shown in the video
4. Expression

   $$10 \bmod 3 = 7 \bmod 3$$
   Explanation

   State the initial symmetric relationship.

   Justification

   Both sides equal 1.

   Shown in the video
5. Expression

   $$7 \bmod 3 = 10 \bmod 3$$
   Explanation

   Flip the relationship to show symmetry.

   Justification

   Equality is symmetric.

   Shown in the video

The relation is symmetric because if 10 mod 3 equals 7 mod 3, then 7 mod 3 also equals 10 mod 3.

1. Audio
   Observation

   Speaker explains the transitive property using the numbers 10, 7, and 13 modulo 3.

2. Formula
   Observation

   Board shows S: 10 mod $3 = 7$ mod 3 => 7 mod $3 = 10$ mod 3, and T: 13 mod 3.

1. Expression

   $$10 \pmod 3 = 7 \pmod 3$$
   Explanation

   First premise given in the example.

   Justification

   Stated by the speaker and written on the board under 'S'.

   Shown in the video
2. Expression

   $$7 \pmod 3 = 13 \pmod 3$$
   Explanation

   Second premise given in the example.

   Justification

   Stated by the speaker.

   Shown in the video
3. Expression

   $$10 \pmod 3 = 13 \pmod 3$$
   Explanation

   Conclusion derived from the two premises.

   Justification

   Application of the transitive property.

   Shown in the video

The relation of having equal remainders modulo 3 is transitive.

1. Audio
   Observation

   The lecturer repeatedly uses the number 3 to test the three properties of equality.

Show, using equality of numbers, how the three properties of an equivalence relation appear.

1. Ordinary equality on numbers

2. Concrete example value 3

Demonstrate reflexivity, symmetry, and transitivity for equality.

1. Expression

   $$3 = 3$$
   Explanation

   This shows that a number is equal to itself.

   Justification

   Reflexivity.

   Shown in the video
2. Expression

   $$3 = 3 \text{ remains true after swapping sides}$$
   Explanation

   Reordering the two sides does not change the truth of the statement.

   Justification

   Symmetry.

   Shown in the video
3. Expression

   $$3 = 3 \text{ and } 3 = 3 \Rightarrow 3 = 3$$
   Explanation

   Chaining the equalities leads back to the same conclusion.

   Justification

   Transitivity.

   Shown in the video

Equality satisfies all three properties and therefore serves as an example of an equivalence relation.

The lecturer explicitly concludes: "So equality is our notion of an equivalence relation."

1. Formula
   Observation

   Handwritten "10 mod 3" appears on screen.

2. Audio
   Observation

   Lecturer says the value is 1 and describes it as the remainder from division.

1. The broader claim that modular equality is an equivalence relation is introduced but not fully worked out before the clip ends.

Evaluate the modular expression 10 mod 3 as the start of a second example of equivalence.

1. Expression 10 mod 3

2. mod means remainder after division

Find the value of 10 mod 3.

1. Expression

   $$10 \bmod 3$$
   Explanation

   Write the modular expression.

   Justification

   Given example.

   Shown in the video
2. Expression

   $$10 \bmod 3 = 1$$
   Explanation

   State the remainder.

   Justification

   Division of 10 by 3 leaves remainder 1.

   Shown in the video

1

The lecturer identifies 1 as the remainder when 10 is divided by 3.

1. Formula
   Observation

   The entire segment uses numbers like 10, 7, and 13 with modulo 3 to illustrate the properties.

Demonstrate that the relation 'a mod $3 = b$ mod 3' is an equivalence relation by checking its reflexive, symmetric, and transitive properties.

1. The relation is defined as a ~ b if and only if a mod $3 = b$ mod 3.

2. Test numbers include 10, 7, and 13.

Verify the three properties of an equivalence relation for the given relation.

1. Expression

   $$10 \bmod 3 = 10 \bmod 3$$
   Explanation

   Check reflexivity using the number 10.

   Justification

   A number is always equal to itself.

   Shown in the video
2. Expression

   $$10 \bmod 3 = 7 \bmod 3 \implies 7 \bmod 3 = 10 \bmod 3$$
   Explanation

   Check symmetry using the numbers 10 and 7.

   Justification

   Both 10 mod 3 and 7 mod 3 equal 1, so they are equivalent. Flipping the order maintains equality.

   Shown in the video
3. Expression

   $$13 \bmod 3 = 1$$
   Explanation

   Introduce a third number, 13, for the transitivity check.

   Justification

   $13 = 4 * 3 + 1$, so its remainder is 1.

   Shown in the video

The segment successfully demonstrates reflexivity and symmetry. It sets up the transitivity check but does not complete the full proof within the provided time.

The numerical values (10 mod $3 = 1$, 7 mod $3 = 1$, 13 mod $3 = 1$) confirm that all tested numbers belong to the same equivalence class under modulo 3.

1. Audio
   Observation

   Speaker uses 10, 7, and 13 to show transitivity.

2. Formula
   Observation

   Handwritten equations on the board.

Show that the relation 'has the same remainder modulo 3' is transitive using specific numbers.

1. $10 \pmod 3 = 7 \pmod 3$

2. $7 \pmod 3 = 13 \pmod 3$

Prove that $10 \pmod 3 = 13 \pmod 3$.

1. Expression

   $$10 \pmod 3 = 1$$
   Explanation

   Calculate the remainder of 10 divided by 3.

   Justification

   Arithmetic calculation.

   Derived from the video
2. Expression

   $$7 \pmod 3 = 1$$
   Explanation

   Calculate the remainder of 7 divided by 3.

   Justification

   Arithmetic calculation.

   Derived from the video
3. Expression

   $$13 \pmod 3 = 1$$
   Explanation

   Calculate the remainder of 13 divided by 3.

   Justification

   Arithmetic calculation.

   Derived from the video
4. Expression

   $$10 \pmod 3 = 13 \pmod 3$$
   Explanation

   Since both sides equal 1, they are equal to each other.

   Justification

   Substitution based on previous steps.

   Derived from the video

$10 \pmod 3 = 13 \pmod 3$

All three numbers (10, 7, 13) leave a remainder of 1 when divided by 3, confirming the equality.

1. Diagram
   Observation

   A cyan-bordered black text box remains in the upper-right throughout the clip.

2. Caption evidence
   Observation

   It reads: "An equivalence relation on a set is a relation that is" with bullets "Reflexive", "Symmetric", "Transitive".

1. Cyan-bordered text box

2. Title phrase "equivalence relation" highlighted in cyan

3. Three bullet points

1. No visible change in the box content during the clip.

1. The definition and three-property list remain continuously visible.

The visual anchor defines the topic and keeps the three required properties in view while the lecturer gives examples.

1. Animation
   Observation

   The lecturer turns toward the board and writes with a pink marker.

2. Formula
   Observation

   The handwritten expression becomes legible as "10 mod 3".

1. Pink handwritten text "10 mod 3"

2. Lecturer's writing hand and marker

1. Before 100s, no handwritten example is present in the main board area.

2. Around 100s the lecturer begins writing.

3. By roughly 110s the full expression "10 mod 3" is visible.

1. The upper-right definition box stays unchanged while the new example is added.

The animation shifts from abstract definition to a concrete second example in modular arithmetic.

1. Animation
   Observation

   The lecturer repeatedly raises fingers, opens palms, and points upward while speaking.

1. Exact one-to-one mapping of each gesture to a specific word is approximate.

1. Lecturer's hands

2. Markers held in hand

1. Early gestures coincide with listing the three properties.

2. Later open-palm and pointing gestures accompany the equality example and transition to modular arithmetic.

1. The speaker remains against a black background with the definition box fixed on screen.

The gestures visually reinforce enumeration of properties and emphasis during the examples.

1. Animation
   Observation

   The speaker writes mathematical expressions on a transparent board while standing in front of a black background.

1. Speaker

2. Transparent board

3. Marker

4. Text box defining equivalence relation

1. The speaker sequentially writes examples for Reflexive (R), Symmetric (S), and Transitive (T) properties.

2. Numbers like 10, 7, and 13 are introduced and their modulo 3 values are calculated as 1.

1. The definition of an equivalence relation remains visible in the top right corner throughout the segment.

The visual progression mirrors the logical structure of proving a relation is an equivalence relation by addressing each required property one by one.

1. Animation
   Observation

   The speaker points to different parts of the handwritten equations on the blackboard while explaining.

1. Handwritten equations for R, S, and T properties

2. Typed definition box

1. Speaker's hand moves to point at specific equations corresponding to the spoken explanation.

1. The text on the board remains static throughout the clip.

The visual layout separates the general definition (typed box) from specific examples (handwritten), aiding comprehension.

1. Audio
   Observation

   Lecturer says an equivalence relation captures the essence of what equality should be "but in a slightly more generalized concept," then introduces modular arithmetic as another example.

One might assume that only standard numerical equality counts as an equivalence relation.

The video explicitly broadens the idea: equivalence relation generalizes the behavior of equality, and modular arithmetic provides another instance.

1. Audio
   Observation

   Now, I haven't really formally proven it, I just did a quick example with some numbers...

Believing that showing a single numerical example constitutes a formal mathematical proof for all cases.

The speaker explicitly states that the demonstration was just a quick example to illustrate the idea, not a formal proof.

1. Audio
   Observation

   After defining equivalence relation, the lecturer immediately uses equality as the first example and checks the three properties.

The abstract definition is applied to ordinary equality to show a familiar case satisfying all three properties.

1. Audio
   Observation

   The lecturer says the generalized nature of equivalence relations will become apparent in the second example, modular arithmetic.

Modular arithmetic is presented as a broader, less obvious example showing that equivalence is not limited to ordinary equality.

1. Caption evidence
   Observation

   The definition box lists Reflexive, Symmetric, Transitive as the contents of the definition.

The concept of equivalence relation contains the three named properties as its defining components.

1. Diagram
   Observation

   The text box lists Reflexive, Symmetric, and Transitive as components of an equivalence relation.

The definition of an equivalence relation contains the requirement of being reflexive.

1. Diagram
   Observation

   The text box lists Reflexive, Symmetric, and Transitive as components of an equivalence relation.

The definition of an equivalence relation contains the requirement of being symmetric.

1. Diagram
   Observation

   The text box lists Reflexive, Symmetric, and Transitive as components of an equivalence relation.

The definition of an equivalence relation contains the requirement of being transitive.

1. Caption evidence
   Observation

   Definition lists Reflexive, Symmetric, Transitive.

The concept of an equivalence relation contains the requirement of transitivity, which is then illustrated by the example.

1. Caption evidence
   Observation

   Definition box directly answers this question.

1. Definition of equivalence relation

1. Audio
   Observation

   The lecturer checks reflexivity, symmetry, and transitivity for equality using $3 = 3$ examples.

1. Equality as a relation on numbers
2. Checking that equality satisfies the three equivalence properties

1. Formula
   Observation

   Handwritten "10 mod 3".

2. Audio
   Observation

   Lecturer explains it as the remainder when 10 is divided by 3.

1. Introduction to modular arithmetic as a second example
2. Modular arithmetic example: 10 mod 3

1. Caption evidence
   Observation

   The three bullets name the properties.

2. Audio
   Observation

   The equality example demonstrates each one in order.

1. Definition of equivalence relation
2. Checking that equality satisfies the three equivalence properties

1. Audio
   Observation

   Speaker explains the reflexive point using an example.

1. Reflexive Property in Modular Arithmetic

1. Audio
   Observation

   Speaker demonstrates the symmetric property by flipping an equation.

1. Symmetric Property in Modular Arithmetic

1. Diagram
   Observation

   Text box provides the definition.

1. Definition of Equivalence Relation

1. Audio
   Observation

   Explanation of transitivity using modulo 3.

1. Transitivity in Modular Arithmetic
2. Transitivity Example with Modulo 3