knowLittle said:Nevermind, you are right. The definition of same cardinality or numerically equivalency doesn't restrict finite from infinite sets.
A bijective relationship is a one-to-one correspondence between two sets, meaning that each element in one set is paired with exactly one element in the other set. This is also known as a "one-to-one and onto" relationship.
To prove a bijective relationship between sets, we must show that there is a one-to-one correspondence between the elements of the two sets. This can be done by showing that for each element in one set, there is a unique corresponding element in the other set, and vice versa.
Two sets have the same cardinality if there is a bijective relationship between them. This means that they have the same number of elements, and each element in one set corresponds to exactly one element in the other set.
To prove a bijective relationship between sets with infinite cardinalities, we can use the Cantor-Bernstein-Schröder theorem. This theorem states that if there are injections (one-to-one mappings) from one set to another and vice versa, then there is a bijective relationship between the two sets.
Yes, we can prove a bijective relationship between the sets (0,1) and (0,2). This can be done by showing that there is a one-to-one correspondence between the elements of the two sets. For example, we can map the element 0.5 from (0,1) to the element 1 from (0,2), and vice versa. This shows that every element in one set has a unique corresponding element in the other set, thus proving a bijective relationship.