Can a relation be both one-to-one and onto?

📌 CONCEPT: A relation can be both one-to-one and onto, but it is not a function. A one-to-one relation is a relation where each element of the domain is mapped to a unique element in the codomain, while an onto relation is a relation where every element in the codomain is mapped to by at least one element in the domain.

📐 RULE / FORMULA: To check if a relation is one-to-one, we need to ensure that no two elements in the domain map to the same element in the codomain. To check if a relation is onto, we need to ensure that every element in the codomain is mapped to by at least one element in the domain.

💡 WORKED EXAMPLE: Let's consider the relation R = {(1, 4), (2, 5), (3, 5)} from set A = {1, 2, 3} to set B = {4, 5}. This relation is one-to-one because each element in the domain maps to a unique element in the codomain. However, it is not onto because the element 4 in the codomain is not mapped to by any element in the domain.

⚠️ COMMON MISTAKE: Students often confuse the concepts of one-to-one and onto relations, assuming that a relation can be both one-to-one and onto only if it is a function. However, this is not necessarily true, as the relation R = {(1, 4), (2, 5), (3, 5)} demonstrates.