An empty relation and a universal relation are two distinct concepts in mathematics, particularly in the context of set theory and relations.
-
Empty Relation: An empty relation is a relation between sets where there are no elements in the relation. In other words, it is a set of ordered pairs with no elements. For example, the empty set ∅ is an empty relation.
-
Universal Relation: A universal relation, on the other hand, is a relation where every possible pair of elements from the sets involved is related. In set theory, this is often represented by the Cartesian product of the sets involved. For example, if A and B are sets, the universal relation on A and B is A×B, which contains all possible ordered pairs from A to B.
No, an empty relation cannot be a universal relation.
This is because:
- An empty relation contains no elements or ordered pairs.
- A universal relation contains all possible ordered pairs between the sets involved.
Since the empty relation contains no elements, it cannot contain all possible ordered pairs. In contrast, a universal relation contains all possible ordered pairs, making it fundamentally different from an empty relation.
No, a universal relation cannot be an empty relation.
A universal relation, as mentioned before, contains all possible ordered pairs between the sets involved. In set theory, if A and B are sets, the universal relation on A and B is represented by A×B, which contains all possible ordered pairs from A to B.
An empty relation, on the other hand, contains no elements or ordered pairs.
Since a universal relation contains all possible ordered pairs, it cannot be empty. It always contains at least one ordered pair, namely, the Cartesian product of the sets involved.
