How does the video use the less-than relation on integers to illustrate order sensitivity?

No, a relation does not have to contain every possible ordered pair. By definition, a relation is a subset of the Cartesian product $A \times B$, meaning it can include only some of the ordered pairs.

Conditions: A and B are sets.; R is a relation between A and B.

Ordered pairs are used because the order of the elements matters in a relation. The first component of the pair is related to the second component, and swapping them can change whether the relationship holds.

Conditions: The relation is between two sets A and B.; The relationship is directional or asymmetric.

In the definition of a relation, capital letters A and B denote the sets themselves, while lowercase letters a and b denote individual elements belonging to those sets. The ordered pair (a, b) represents a specific element of the Cartesian product $A \times B$, where a is an element of A and b is an element of B.

Conditions: The context is the formal definition of a relation R between sets A and B.; The notation (a, b) ∈ $A \times B$ is used.

Yes, elements of the sets can be left out of a relation. A relation is a subset of the Cartesian product, so it does not need to include every possible ordered pair.

Conditions: A relation R is defined between sets A and B.; R is a subset of $A \times B$.

In a relation diagram, an arrow drawn from an element in the first set to an element in the second set means that the corresponding ordered pair belongs to the relation. It visually encodes the membership of a specific pair in the subset of the Cartesian product.

Conditions: The diagram shows two sets, typically represented as ovals or circles.; An arrow connects an element from the first set to an element from the second set.