Why are ordered pairs used in the definition of a relation between two sets?

The video tests the less-than relation with two reversed ordered pairs: (2, 5) and (5, 2). It shows that $2 < 5$ is true, so the pair (2, 5) belongs to the relation.

Conditions: The relation is the less-than symbol (<) on integers.; The pairs tested are (2, 5) and (5, 2).

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.

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.