How does the humans-and-pets example illustrate that a relation is a subset of the Cartesian product?
Conditions
- The sets are humans {, , , } and pets {, , , }.
- The relation is ownership.
Reasoning, step by step
- List the elements of the humans set and the pets set.
- Conceptualize the Cartesian product as all possible ordered pairs (human, pet).
- Identify the specific ordered pairs that satisfy the ownership relation (e.g., (, ), (, )).
- Note the ordered pairs that do not satisfy the relation (e.g., (, )).
- Conclude that the relation consists only of the satisfying pairs, making it a subset of the Cartesian product.
Example
The video draws arrows for specific ownerships: to , to , and to . It leaves , , and unconnected. The lecturer states that some pairings were in the relation and some were not, illustrating that it is not all possible pairings.
Common misconceptions
- Assuming the relation includes all humans and all pets.
- Thinking the Cartesian product is the same as the relation.
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Related questions
The video tests the less-than relation with two reversed ordered pairs: (2, 5) and (5, 2). It shows that 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 , 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 , 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) ∈ 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 .
Answers are generated from source material and independently checked. Consult the original video or creator if something is unclear.