Question

Consider the following five relations on the set A={1,2,3,4}:

R1={(1,1),(1,2),(2,3),(1,3),(4,4)}
R2={(1,1)(1,2),(2,1),(2,2),(3,3),(4,4)}
R3={(1,3),(2,1)}
R4=∅, the empty relation
R5=A×A, the universal relation

Answer 1

The question deals with different types of relations defined on a set A in the context of set theory and discrete mathematics.

Step-by-step explanation:

The student is asking about the concept of relations in mathematics, specifically within the context of set theory. The relations are defined on a set A, and the student has presented five different relations as examples: R1, R2, R3, R4, and R5. To clarify, R1 includes ordered pairs where the first element is related to the second, R2 is similar but includes both (1, 2) and (2, 1), suggesting a symmetry. R3 contains only two ordered pairs without any specific property readily apparent. R4 is the empty relation with no elements, while R5, the universal relation, contains all possible ordered pairs from set A.

Understanding these different types of relations is crucial for grasping foundational concepts in set theory and discrete mathematics. These concepts have applications in computer science, logic, and advanced mathematics. Relations can exhibit certain properties such as reflexivity, symmetry, and transitivity, which are important for understanding the structure and characteristics of a given relation.