Relations between two sets | Definition + First Examples

1. Caption evidence
   Observation

   Top-left slide title reads "Relations:".

Relations:

Section heading naming the topic of relations.

Topic label on the slide

1. Caption evidence
   Observation

   Slide text shows "Ex: $a < b$".

2. Audio
   Observation

   The speaker explains comparing two different integers by asking whether one integer is less than the other.

a, b

Variables standing for two integers being compared by the less-than relation.

Integers

1. Caption evidence
   Observation

   Slide text uses the symbol in "$a < b$".

2. Audio
   Observation

   The speaker calls it "the less than symbol" and uses it to compare integers.

<

Binary relation expressing that the first argument is less than the second argument.

Used here on integers

1. Animation
   Observation

   Handwritten example "$2 < 5$" appears under the bullet about some pairs having the relationship.

2. Audio
   Observation

   The speaker says it is true that 2 is less than 5 and identifies the ordered pair (2, 5) as having this relationship.

$2 < 5$

Concrete true instance of the less-than relation on integers.

Integers

1. Audio
   Observation

   The speaker explicitly says "this particular pairing, 2 comma 5, that ordered pair, has this particular relationship."

(2, 5)

Ordered pair whose first entry is 2 and second entry is 5; used to represent an instance of a relation.

Pairs of integers

1. Animation
   Observation

   Handwritten example "$5 < 2$" is written and then crossed out with a slash.

2. Audio
   Observation

   The speaker says "is 5 less than 2? Well, this is not the case" and adds a not symbol through it.

$5 \not< 2$

Concrete false instance showing that the ordered pair (5, 2) does not satisfy the less-than relation.

Integers

1. Animation
   Observation

   Left orange oval is populated with labels $h_1$, $h_2$, $h_3$, $h_4$.

2. Audio
   Observation

   The speaker says this set contains a bunch of different humans and names human 1 through human 4.

$h_1$, $h_2$, $h_3$, $h_4$

Labels for elements of the left set, each representing a human.

Elements of the humans set

1. Animation
   Observation

   Right orange oval is populated with labels $d_1$, $c_1$, $d_2$, $m_1$.

2. Audio
   Observation

   The speaker says the right set contains pets: a dog $d_1$, a cat $c_1$, another dog $d_2$, and a monkey $m_1$.

$d_1$, $c_1$, $d_2$, $m_1$

Labels for elements of the right set, each representing a pet.

Elements of the pets set

1. Audio
   Observation

   The speaker says, "we will often visually depict a set as just a big circle, and then you just write the elements inside of the circle."

2. Animation
   Observation

   Two large ovals are drawn and filled with element labels.

big circle / oval

Visual convention for depicting a set, with elements written inside.

Diagrammatic notation for sets

1. Diagram
   Observation

   Left orange oval contains four labeled elements $h_1$, $h_2$, $h_3$, $h_4$.

2. Audio
   Observation

   The speaker refers to these as humans in the ownership example.

$h_1$, $h_2$, $h_3$, $h_4$

Elements of the first set, interpreted as humans.

Members of the left set in the example diagram.

1. Diagram
   Observation

   Right orange oval contains four labeled elements $d_1$, $c_1$, $d_2$, $m_1$.

2. Audio
   Observation

   The speaker refers to these as pets and names cat, dog, and monkey examples.

$d_1$, $c_1$, $d_2$, $m_1$

Elements of the second set, interpreted as pets.

Members of the right set in the example diagram.

1. Animation
   Observation

   Pink arrows are drawn from $h_1$ to $c_1$, from $h_2$ to $d_1$, and from $h_3$ to $d_1$.

2. Audio
   Observation

   The speaker says one way to depict the relationship is by drawing an arrow that connects these elements.

arrow from a to b

Visual depiction that the ordered pair (a,b) belongs to the relation.

Used on the example diagram between the two sets.

1. Audio
   Observation

   The speaker says mathematics often wants to relate one item or object to another and to say there is some comparison or relationship between them.

2. Caption evidence
   Observation

   Slide title is "Relations:".

A relation is introduced informally as a way to connect or compare one mathematical object with another. The video presents this as the motivating idea before giving examples.

1. Applies when comparing or linking two objects.

1. Caption evidence
   Observation

   Slide text shows "Ex: $a < b$" and bullet "Compares two integers".

2. Audio
   Observation

   The speaker says one familiar relation is the less-than symbol, used to compare two different integers.

The expression $a < b$ is presented as an example of a relation: it compares two integers and tells whether the first is less than the second.

1. The example is stated for integers.

2. It compares two objects in an ordered way.

1. Informal idea of a relation

1. Caption evidence
   Observation

   Bullet text reads "Some pairs have this relationship".

2. Animation
   Observation

   Handwritten example "$2 < 5$" appears.

3. Audio
   Observation

   The speaker says it is true that 2 is less than 5 and that the ordered pair (2, 5) has this relationship.

A relation does not hold for every possible pair. In the less-than example, the ordered pair $(2,5)$ satisfies the relation because $2<5$ is true.

1. The pair must be ordered.

2. Truth of the statement determines membership in the relation.

1. Less-than as a relation on integers

1. Caption evidence
   Observation

   Bullet text reads "Some pairs don't".

2. Animation
   Observation

   Handwritten "$5 < 2$" is crossed out.

3. Audio
   Observation

   The speaker says if we go the other way around, 5 is not less than 2, and marks it with a not symbol.

The same relation can fail for other ordered pairs. The reversed pair $(5,2)$ does not satisfy $a<b$, so the video marks $5<2$ as false.

1. Order matters in the pair.

2. This example contrasts with (2,5).

1. Less-than as a relation on integers
2. Some ordered pairs satisfy a relation

1. Audio
   Observation

   The speaker says sets are often visually depicted as a big circle with elements written inside.

2. Animation
   Observation

   A large orange oval is drawn on the left side of the board.

The video introduces a diagrammatic convention: draw a large circle or oval to represent a set, then place the elements inside it.

1. Used for visual representation rather than formal definition.

1. Animation
   Observation

   Left oval contains $h_1$, $h_2$, $h_3$, $h_4$; right oval contains $d_1$, $c_1$, $d_2$, $m_1$.

2. Audio
   Observation

   The speaker describes one set as humans and the other as pets, naming dog, cat, another dog, and monkey.

A second example begins by drawing two separate sets: a humans set with elements $h_1,h_2,h_3,h_4$ and a pets set with elements $d_1,c_1,d_2,m_1$. This prepares the idea of relating elements from one set to elements of another set.

1. The two sets are distinct.

2. The clip stops before any arrows or explicit relation between them are shown.

1. Visual depiction of a set

1. Audio
   Observation

   The speaker says there could be some relationships and proposes making the relation be ownership: the human owns the pet.

2. Diagram
   Observation

   Two orange ovals with elements $h_1$,$h_2$,$h_3$,$h_4$ and $d_1$,$c_1$,$d_2$,$m_1$ are already on the board.

3. Animation
   Observation

   Pink arrows are added from $h_1$ to $c_1$, $h_2$ to $d_1$, and $h_3$ to $d_1$.

The clip first presents a relation informally as a connection or ownership link between an element of one set and an element of another set. The visual convention is an arrow from the first-set element to the second-set element.

1. The example uses two distinct sets.

2. Arrows indicate which cross-set pairs are related.

1. Audio
   Observation

   The speaker explicitly names the pairs $h_2 d_1$ and $h_3 d_1$, then writes ($h_1$,$d_1$) and explains its meaning.

2. Formula
   Observation

   The board shows ($h_1$,$d_1$) means human $h_1$ owns $d_1$.

The video translates the arrow picture into ordered-pair language: if an arrow goes from a human to a pet, then the corresponding ordered pair belongs to the relation. It also stresses that the order matters because the first component is the human and the second component is the pet.

1. The first coordinate comes from the first set.

2. The second coordinate comes from the second set.

1. Informal idea of a relation as a connection between elements of two sets

1. Caption evidence
   Observation

   Red-bordered box reads: Defn: A Relation R between A and B Is a subset of $A \times B$.

2. Audio
   Observation

   The speaker says, 'So our formal definition is this,' and begins naming the relation R.

1. The spoken sentence is cut off before the full verbal explanation of the displayed definition is completed.

After the example, the clip gives the standard formal definition: a relation R between sets A and B is any subset of the Cartesian product $A \times B$. In other words, a relation is a chosen collection of ordered pairs drawn from all possible pairs with first entry in A and second entry in B.

1. A and B are sets.

2. R consists of ordered pairs whose first component is in A and second component is in B.

1. Relation membership expressed by ordered pairs

1. Caption evidence
   Observation

   Persistent top-left overlay: "Defn: A Relation R between A and B Is a subset of $A \times B$".

2. Audio
   Observation

   The lecturer explains that a relation is a relationship between two different sets, A and B, and that it is a subset of A cross B.

3. Formula
   Observation

   Around 48 seconds the board displays "(a,b) ∈ $A\times B$" to unpack what kind of objects are in the subset.

The clip defines a relation R between two sets A and B as a subset of the Cartesian product $A \times B$. The lecturer unpacks this by saying that the elements being collected are ordered pairs, and that lowercase a and b denote elements while capital A and B denote sets.

1. A and B are sets.

2. R is a collection of ordered pairs belonging to $A \times B$.

3. The video does not state whether A and B must be distinct; it only says "between two different sets" in the spoken explanation.

1. Ordered pairs as elements of a Cartesian product
2. Relation as a selected collection inside $A \times B$

1. Audio
   Observation

   The lecturer says ordered pairs are elements of the Cartesian product, A cross B.

2. Formula
   Observation

   The board shows "(a,b) ∈ $A\times B$" after writing "ordered pairs".

3. Audio
   Observation

   He says lowercase a and b denote elements, while capital A and B denote sets.

The lecture treats $A \times B$ as the Cartesian product whose elements are ordered pairs. The displayed membership statement makes the role of (a,b) explicit: an ordered pair is an element of $A \times B$.

1. a is an element of A.

2. b is an element of B.

3. The clip does not separately define ordered-pair equality or the full formal construction of $A \times B$.

1. Audio
   Observation

   The lecturer says a relation is some subset, some collection of ordered pairs within the Cartesian product, but not necessarily the entirety.

2. Audio
   Observation

   He adds that taking all possible pairings could be a trivial relationship.

1. The exact formal notation for proper versus improper subset is not discussed in the clip.

To understand the definition, the lecturer interprets "subset of $A \times B$" as choosing some collection of ordered pairs from the full Cartesian product. The chosen collection may be smaller than all possible pairings, although the full set of pairings is mentioned as a possible trivial relation.

1. The relation consists of ordered pairs from $A \times B$.

2. It need not contain every element of $A \times B$.

3. The full Cartesian product is described as a possible trivial relationship, not as the only allowed case.

1. Definition of a relation between two sets
2. Ordered pairs as elements of a Cartesian product

1. Audio
   Observation

   The speaker states that in mathematics we often want to relate one item or object to another and say there is some comparison or relationship between them.

A relation expresses a comparison or connection between one object and another.

1. There are at least two objects being considered.

Informal universal motivation: very often in mathematics.

1. Caption evidence
   Observation

   "Ex: $a < b$" with bullet "Compares two integers".

2. Audio
   Observation

   The speaker says we can ask whether one integer is less than the other.

For integers $a$ and $b$, the expression $a<b$ tells whether one integer is less than the other, thereby giving a relationship between them.

1. $a$ and $b$ are integers.

For the displayed example involving two integers.

1. Audio
   Observation

   The speaker says the ordered pair (2, 5) has this particular relationship.

2. Animation
   Observation

   The board shows "$2 < 5$".

Because $2<5$ is true, the ordered pair $(2,5)$ has the less-than relationship.

1. The relation under discussion is $<$ on integers.

2. The pair is taken in the order shown.

Specific example.

1. Audio
   Observation

   The speaker says if we go the other way around, 5 is not less than 2.

2. Animation
   Observation

   "$5 < 2$" is written and then crossed out.

The reversed ordered pair $(5,2)$ does not satisfy the less-than relation, since $5<2$ is false.

1. The relation is $<$ on integers.

2. The pair order is reversed relative to (2,5).

Specific counterexample within the same relation.

1. Audio
   Observation

   The speaker says that among all ordered pairs of integers, sometimes those pairs have this relationship and sometimes they do not.

1. The speaker does not formally define the relation as a set of ordered pairs in this clip.

Among all ordered pairs of integers, some satisfy the less-than relation and some do not.

1. Consider ordered pairs of integers.

2. Use the relation $<$.

Over all ordered pairs of integers.

1. Audio
   Observation

   The speaker says sets are often visually depicted as a big circle with elements written inside.

A common visual convention is to depict a set as a large circle and write its elements inside the circle.

1. Using a diagrammatic representation of sets.

General convention stated by the speaker.

1. Caption evidence
   Observation

   Bullets state "Some pairs have this relationship" and "Some pairs don't".

2. Animation
   Observation

   Examples "$2 < 5$" and crossed-out "$5 < 2$" are written.

3. Audio
   Observation

   The speaker verbally contrasts the true and false cases.

1. Expression

   $$a<b$$
   Explanation

   Start with the general relation comparing two integers.

   Justification

   Given by the slide example and the speaker's introduction of the less-than symbol.

   Shown in the video
2. Expression

   $$2<5$$
   Explanation

   Instantiate the relation with the ordered pair (2,5).

   Justification

   Direct substitution of concrete integers into the displayed relation.

   Shown in the video
3. Expression

   $$(2,5)\text{ has the relationship}$$
   Explanation

   Conclude that this ordered pair belongs to the relation because the statement is true.

   Justification

   Stated explicitly by the speaker.

   Shown in the video
4. Expression

   $$5<2$$
   Explanation

   Now test the reversed ordered pair (5,2).

   Justification

   The speaker says to go the other way around.

   Shown in the video
5. Expression

   $$5\not<2$$
   Explanation

   Mark the reversed statement as false.

   Justification

   The speaker says this is not the case and crosses out the inequality on the board.

   Shown in the video

The less-than relation distinguishes ordered pairs: (2,5) satisfies it, while (5,2) does not.

1. Animation
   Observation

   Two ovals are drawn and populated with element labels.

2. Audio
   Observation

   The speaker explains the circle convention and then defines one set of humans and one set of pets.

1. The actual relation between the two sets is not yet shown in this clip.

1. Expression

   $$\text{draw a big circle for a set}$$
   Explanation

   Introduce the visual convention for representing sets.

   Justification

   Explicitly stated by the speaker.

   Shown in the video
2. Expression

   $$\{h_1,h_2,h_3,h_4\}$$
   Explanation

   Populate the left set with four human elements.

   Justification

   Shown by handwritten labels and described in audio.

   Shown in the video
3. Expression

   $$\{d_1,c_1,d_2,m_1\}$$
   Explanation

   Populate the right set with four pet elements.

   Justification

   Shown by handwritten labels and described in audio.

   Shown in the video
4. Expression

   $$\text{two different sets are now available for comparison}$$
   Explanation

   Prepare to relate elements from the humans set to elements from the pets set.

   Justification

   The speaker ends by saying to think about these two different sets.

   Shown in the video

The clip establishes two distinct sets, one of humans and one of pets, as the domain and codomain for a forthcoming relation.

1. Animation
   Observation

   Arrows are drawn one by one from $h_1$ to $c_1$, then $h_2$ to $d_1$, then $h_3$ to $d_1$.

2. Audio
   Observation

   The speaker narrates each ownership example and later says both pairs $h_2 d_1$ and $h_3 d_1$ have the relationship.

3. Formula
   Observation

   The board later writes ($h_1$,$d_1$) means human $h_1$ owns $d_1$.

1. Expression

   $$$$
   Explanation

   Start with two separate sets: humans on the left and pets on the right.

   Justification

   This is the initial board state visible at the beginning of the clip.

   Shown in the video
2. Expression

   $$$$
   Explanation

   Draw an arrow from $h_1$ to $c_1$ to represent that the first human owns the cat.

   Justification

   The speaker explicitly says the first human owns the cat while drawing the arrow.

   Shown in the video
3. Expression

   $$$$
   Explanation

   Draw an arrow from $h_2$ to $d_1$ to represent that the second human owns the first dog.

   Justification

   The speaker explicitly says the second human owns this dog while drawing the arrow.

   Shown in the video
4. Expression

   $$$$
   Explanation

   Draw an arrow from $h_3$ to $d_1$ to show that the third human also owns the same dog.

   Justification

   The speaker says the third and second humans co-own the first dog and names the pair $h_3 d_1$.

   Shown in the video
5. Expression

   $$(h_2,d_1),\ (h_3,d_1)$$
   Explanation

   State the co-ownership situation as two ordered pairs in the relation.

   Justification

   The speaker directly names both pairs as having the human-owns-pet relationship.

   Shown in the video
6. Expression

   $$(h_1,d_1)\text{ means human }h_1\text{ owns }d_1$$
   Explanation

   Use one ordered pair to define the interpretation rule for the whole picture.

   Justification

   The speaker writes this exact notation and explains its meaning aloud.

   Shown in the video

The arrow diagram is just a visual encoding of a set of ordered pairs, and the clip uses that bridge to motivate the formal definition.

1. Caption evidence
   Observation

   The overlay gives the compact definition "A Relation R between A and B Is a subset of $A \times B$".

2. Audio
   Observation

   The lecturer verbally expands the definition by referring to ordered pairs and the Cartesian product.

3. Formula
   Observation

   The board writes "ordered pairs" and then "(a,b) ∈ $A\times B$".

1. Expression

   $$R \text{ is a relation between } A \text{ and } B$$
   Explanation

   The starting point is the displayed definition naming a relation R between two sets A and B.

   Justification

   Directly shown in the top-left overlay and spoken by the lecturer.

   Shown in the video
2. Expression

   $$R \subseteq A \times B$$
   Explanation

   The definition says that R is a subset of the Cartesian product $A \times B$.

   Justification

   Stated in the overlay as "Is a subset of $A \times B$" and repeated in the audio.

   Shown in the video
3. Expression

   $$\text{elements of } A \times B \text{ are ordered pairs}$$
   Explanation

   The lecturer clarifies that the objects being collected in the subset are ordered pairs.

   Justification

   Audio: ordered pairs are elements of the Cartesian product A cross B.

   Shown in the video
4. Expression

   $$(a,b) \in A \times B$$
   Explanation

   A representative ordered pair is written on the board to show the form of the elements in the Cartesian product.

   Justification

   Visible formula written during the explanation.

   Shown in the video
5. Expression

   $$a,b \text{ lowercase denote elements; } A,B \text{ uppercase denote sets}$$
   Explanation

   The lecturer distinguishes the notation for elements and sets while explaining the written pair.

   Justification

   Audio explicitly states lowercase letters are elements and capital letters are sets.

   Shown in the video
6. Expression

   $$R \text{ is some collection of ordered pairs within } A \times B$$
   Explanation

   The conclusion is that a relation is not an arbitrary kind of object but a selected set of ordered pairs drawn from the Cartesian product.

   Justification

   Audio summarizes the definition as a subset, a collection of ordered pairs, not necessarily the entirety.

   Shown in the video

A relation R between A and B is understood as a subset of $A \times B$, i.e. a collection of ordered pairs (a,b) drawn from the Cartesian product.

1. Audio
   Observation

   The lecturer refers to a first example where A and B are real numbers or integers and the relationship was less than.

2. Audio
   Observation

   He says the pair 2,5 would be in there, but the pair 5,2 would not.

1. No board formula is written for this example; it is conveyed orally.

1. Expression

   $$A,B \in \{\mathbb{R},\mathbb{Z}\}$$
   Explanation

   The example takes the two sets to be real numbers or integers.

   Justification

   Spoken description of the earlier example.

   Shown in the video
2. Expression

   $$R = \{(a,b): a<b\}$$
   Explanation

   The relation is the collection of ordered pairs satisfying the property "less than."

   Justification

   Audio says the relationship was less than and that the relation consisted of all ordered pairs with that property.

   Derived from the video
3. Expression

   $$(2,5) \in R$$
   Explanation

   The ordered pair with first coordinate 2 and second coordinate 5 satisfies $2<5$, so it belongs to the relation.

   Justification

   Audio explicitly says the pair 2,5 would be in there.

   Shown in the video
4. Expression

   $$(5,2) \notin R$$
   Explanation

   The reversed ordered pair does not satisfy the less-than property, so it is not in the relation.

   Justification

   Audio explicitly says the pair 5,2 would not be in there.

   Shown in the video

For the less-than relation on numbers, membership depends on the ordered pair satisfying the stated property, and reversing the coordinates can change membership.

1. Audio
   Observation

   The lecturer refers to humans and pets examples where some pairings were in the relation and some were not.

2. Audio
   Observation

   He says it is not all possible pairings, though all possible pairings could be a trivial relationship.

1. The specific humans and pets are not named in this clip.

1. Expression

   $$A=\text{humans},\quad B=\text{pets}$$
   Explanation

   The example uses a set of humans and a set of pets as the two underlying sets.

   Justification

   Audio refers to humans and pets examples.

   Shown in the video
2. Expression

   $$R \subsetneq A \times B \text{ in the nontrivial example}$$
   Explanation

   Some human-pet pairings are included and others are excluded, so the relation is a proper selection from the Cartesian product.

   Justification

   Audio says some pairings were in there and some were not, and that it is not all possible pairings.

   Derived from the video
3. Expression

   $$R=A \times B \text{ could be a trivial relationship}$$
   Explanation

   If every possible pairing were included, the relation would be the entire Cartesian product, which the lecturer describes as possibly trivial.

   Justification

   Audio says all possible pairings could be a kind of trivial relationship.

   Shown in the video

A relation can include only some ordered pairs from $A \times B$; including all possible pairings is a special trivial case rather than the usual intended situation.

1. Caption evidence
   Observation

   Slide text: "Ex: $a < b$", "Compares two integers", "Some pairs have this relationship", "Some pairs don't".

2. Animation
   Observation

   Handwritten examples "$2 < 5$" and crossed-out "$5 < 2$".

3. Audio
   Observation

   The speaker explains both the true and false cases and mentions the ordered pair (2,5).

Use the relation $a<b$ to compare integers and identify which ordered pairs satisfy it.

1. $a$ and $b$ are integers.

2. The relation is $<$.

3. Concrete pairs to test include (2,5) and (5,2).

Determine whether each ordered pair has the less-than relationship.

1. Expression

   $$a<b$$
   Explanation

   Write the general relational form for comparing two integers.

   Justification

   Displayed on the slide and introduced verbally.

   Shown in the video
2. Expression

   $$2<5$$
   Explanation

   Test the pair (2,5).

   Justification

   Substitute the specific integers into the relation.

   Shown in the video
3. Expression

   $$\text{true}$$
   Explanation

   Conclude that (2,5) has the relationship.

   Justification

   The speaker states it is a true fact and names the ordered pair (2,5).

   Shown in the video
4. Expression

   $$5<2$$
   Explanation

   Test the reversed pair (5,2).

   Justification

   The speaker explicitly goes the other way around.

   Shown in the video
5. Expression

   $$\text{false}$$
   Explanation

   Conclude that (5,2) does not have the relationship.

   Justification

   The speaker says this is not the case and crosses out the inequality.

   Shown in the video

$(2,5)$ satisfies the relation $a<b$, while $(5,2)$ does not.

The board contrast between "$2 < 5$" and crossed-out "$5 < 2$" matches the spoken explanation.

1. Animation
   Observation

   Left oval labeled with $h_1$, $h_2$, $h_3$, $h_4$; right oval labeled with $d_1$, $c_1$, $d_2$, $m_1$.

2. Audio
   Observation

   The speaker says one set contains humans and the other contains pets, naming dog, cat, another dog, and monkey.

1. No arrows or explicit ownership relation are shown before the clip ends.

Represent two different sets visually as a setup for discussing a relation between them.

1. One set consists of humans.

2. Another set consists of pets.

3. Elements are named $h_1$,$h_2$,$h_3$,$h_4$ and $d_1$,$c_1$,$d_2$,$m_1$.

Draw and label the two sets that may later be related to each other.

1. Expression

   $$\text{draw left oval}$$
   Explanation

   Create the first set container.

   Justification

   The speaker introduces the circle convention for sets.

   Shown in the video
2. Expression

   $$\{h_1,h_2,h_3,h_4\}$$
   Explanation

   Insert four human elements into the left set.

   Justification

   Visible labels and matching narration.

   Shown in the video
3. Expression

   $$\text{draw right oval}$$
   Explanation

   Create the second set container.

   Justification

   The speaker says he will draw a different set.

   Shown in the video
4. Expression

   $$\{d_1,c_1,d_2,m_1\}$$
   Explanation

   Insert four pet elements into the right set.

   Justification

   Visible labels and matching narration identifying dog, cat, another dog, and monkey.

   Shown in the video

The clip produces two labeled sets: a humans set and a pets set, ready for a relation between them.

The final frame shows both ovals fully populated exactly as described in the audio.

1. Diagram
   Observation

   Left set contains $h_1$,$h_2$,$h_3$,$h_4$; right set contains $d_1$,$c_1$,$d_2$,$m_1$.

2. Animation
   Observation

   Pink arrows connect $h_1\to c_1$, $h_2\to d_1$, and $h_3\to d_1$.

3. Audio
   Observation

   The speaker describes ownership, co-ownership, and elements with no arrows.

1. The exact semantic labels of $d_2$ and $m_1$ are inferred from the speaker's mention of another dog and a monkey; the board itself only shows the symbols.

Depict a relation between a set of humans and a set of pets using arrows and ordered pairs.

1. Human set shown as {$h_1$,$h_2$,$h_3$,$h_4$}.

2. Pet set shown as {$d_1$,$c_1$,$d_2$,$m_1$}.

3. Ownership is chosen as the relation.

Show which cross-set pairs are related and interpret the picture as ordered pairs.

1. Expression

   $$h_1 \to c_1$$
   Explanation

   The first human is connected to the cat.

   Justification

   The speaker says the first human owns the cat and draws that arrow.

   Shown in the video
2. Expression

   $$h_2 \to d_1$$
   Explanation

   The second human is connected to the first dog.

   Justification

   The speaker says the second human owns this dog and draws that arrow.

   Shown in the video
3. Expression

   $$h_3 \to d_1$$
   Explanation

   The third human is also connected to the first dog.

   Justification

   The speaker says the third and second humans co-own the first dog and names the pair $h_3 d_1$.

   Shown in the video
4. Expression

   $$h_4\text{ has no outgoing arrow}$$
   Explanation

   The fourth human is left unconnected.

   Justification

   The speaker says maybe this human doesn't own any pets.

   Shown in the video
5. Expression

   $$d_2,\ m_1\text{ have no incoming arrows}$$
   Explanation

   The remaining dog and the monkey are left unconnected.

   Justification

   The speaker says the dog and the monkey don't have any human owners.

   Shown in the video
6. Expression

   $$(h_1,d_1)\text{ means human }h_1\text{ owns }d_1$$
   Explanation

   The clip rewrites the picture in ordered-pair notation.

   Justification

   The speaker writes this notation and explains it verbally.

   Shown in the video

The example relation is visually represented by the arrows $h_1\to c_1$, $h_2\to d_1$, and $h_3\to d_1$, and verbally interpreted as ordered pairs such as ($h_1$,$d_1$), ($h_2$,$d_1$), and ($h_3$,$d_1$) belonging to the ownership relation.

The speaker checks the interpretation by pointing out which elements have arrows and which do not, then restates the meaning of an ordered pair in words.

1. Audio
   Observation

   The lecturer says, in the first example, A and B could be real numbers or integers and the relationship was less than.

2. Audio
   Observation

   He gives the ordered pairs 2,5 and 5,2, saying the first would be in the relation and the second would not.

1. The example is oral only; no corresponding formula is written on the board in this clip.

Describe the relation "less than" on numbers and decide whether the ordered pairs (2,5) and (5,2) belong to it.

1. A and B are real numbers or integers.

2. The relation is the less-than relationship.

Identify which ordered pairs satisfy the relation and therefore belong to the relation.

1. Expression

   $$R=\{(a,b):a<b\}$$
   Explanation

   The relation consists of all ordered pairs whose first coordinate is less than the second coordinate.

   Justification

   Derived from the spoken description that the relationship was less than and consisted of ordered pairs with that property.

   Derived from the video
2. Expression

   $$(2,5)\in R$$
   Explanation

   Since $2<5$, the ordered pair (2,5) has the required property.

   Justification

   Audio explicitly states that the pair 2,5 would be in there.

   Shown in the video
3. Expression

   $$(5,2)\notin R$$
   Explanation

   Since $5<2$ is false, the ordered pair (5,2) does not have the required property.

   Justification

   Audio explicitly states that the pair 5,2 would not be in there.

   Shown in the video

(2,5) is in the less-than relation, while (5,2) is not.

Check the defining inequality $a<b$ for each ordered pair.

1. Audio
   Observation

   The lecturer refers to humans and pets examples where some pairings of humans and pets were in the relation and some were not.

2. Audio
   Observation

   He says it is not all possible pairings, though all possible pairings could be a trivial relationship.

1. No specific human names, pet names, or written pairs are shown in this clip.

Use a relation between humans and pets to illustrate that a relation need not contain every possible pairing from the Cartesian product.

1. One set consists of humans.

2. Another set consists of pets.

3. Some human-pet pairings are included in the relation and some are not.

Show how a relation is a selected subset of possible ordered pairs rather than necessarily the whole Cartesian product.

1. Expression

   $$A=\text{humans},\quad B=\text{pets}$$
   Explanation

   Take the two underlying sets to be humans and pets.

   Justification

   Spoken reference to the humans and pets examples.

   Shown in the video
2. Expression

   $$R\subseteq A\times B$$
   Explanation

   A relation between these sets is a collection of ordered human-pet pairs drawn from the Cartesian product.

   Justification

   Follows from the displayed definition of a relation as a subset of $A \times B$.

   Derived from the video
3. Expression

   $$\text{some }(h,p)\in R,\quad \text{some }(h,p)\notin R$$
   Explanation

   In the example, only certain pairings are included, so the relation is not the full set of all possible human-pet pairs.

   Justification

   Audio says some pairings were in there and some were not.

   Shown in the video
4. Expression

   $$R=A\times B \text{ is possible but trivial}$$
   Explanation

   Including every possible human-pet pairing would make the relation the entire Cartesian product, which the lecturer calls a possible trivial relationship.

   Justification

   Audio says all possible pairings could be a kind of trivial relationship.

   Shown in the video

The humans-and-pets example illustrates that a relation is some subset of $A \times B$, usually not all possible pairings, while the full Cartesian product is a trivial special case.

Compare the chosen relation with the full Cartesian product $A \times B$ and check whether any possible ordered pair is omitted.

1. Caption evidence
   Observation

   The slide begins with only "Relations:" and then adds "Ex: $a < b$" and the bullet "Compares two integers".

2. Caption evidence
   Observation

   Later the bullet "Some pairs have this relationship" appears.

1. Title "Relations:"

2. Example line "Ex: $a < b$"

3. Bullet "Compares two integers"

4. Bullet "Some pairs have this relationship"

1. At about 18 seconds, the example line and first bullet appear.

2. At about 34 seconds, the second bullet appears.

1. The lecturer remains on the right side of the frame.

2. The background stays black and the text stays white.

The visual sequence introduces the topic, then narrows to the concrete less-than example, then states that only some pairs satisfy the relation.

1. Animation
   Observation

   Handwritten orange "$2 < 5$" appears under the bullet about some pairs having the relationship.

2. Animation
   Observation

   Handwritten orange "$5 < 2$" appears lower on the board and is then crossed out with a slash.

1. Handwritten "$2 < 5$"

2. Handwritten "$5 < 2$"

3. Slash through "$5 < 2$"

1. First the true example is written.

2. Then the reversed example is written below it.

3. Finally the reversed example is marked false with a slash.

1. Both examples use the same less-than symbol.

2. The printed slide bullets remain visible above the handwriting.

The animation contrasts an ordered pair that belongs to the relation with one that does not, emphasizing order sensitivity.

1. Animation
   Observation

   A large orange oval is drawn on the left, then filled with $h_1$, $h_2$, $h_3$, $h_4$.

2. Animation
   Observation

   A second orange oval is drawn on the right, then filled with $d_1$, $c_1$, $d_2$, $m_1$.

1. No connecting arrows are present in this clip.

1. Left oval

2. Right oval

3. Labels $h_1$, $h_2$, $h_3$, $h_4$

4. Labels $d_1$, $c_1$, $d_2$, $m_1$

1. Left oval is drawn first.

2. Human labels are added inside the left oval.

3. Right oval is drawn next.

4. Pet labels are added inside the right oval.

1. Both sets are represented by similar orange ovals.

2. The elements remain inside their respective ovals.

The visual establishes two distinct collections of objects, setting up a relation between elements of different sets.

1. Diagram
   Observation

   Two orange ovals are visible with $h_1$,$h_2$,$h_3$,$h_4$ on the left and $d_1$,$c_1$,$d_2$,$m_1$ on the right.

2. Audio
   Observation

   The speaker introduces the idea that there could be some relationships.

1. Left oval with $h_1$,$h_2$,$h_3$,$h_4$

2. Right oval with $d_1$,$c_1$,$d_2$,$m_1$

3. Title 'Relations:'

1. No arrows are present yet.

1. The two sets remain separate before the relation is drawn.

This establishes the domain and codomain-like sets between which a binary relation will be depicted.

1. Animation
   Observation

   Pink arrows are added sequentially from $h_1$ to $c_1$, then $h_2$ to $d_1$, then $h_3$ to $d_1$.

2. Audio
   Observation

   The speaker narrates ownership and co-ownership while drawing.

1. Arrow $h_1\to c_1$

2. Arrow $h_2\to d_1$

3. Arrow $h_3\to d_1$