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Terrell
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why is not always true that if ##\vert A\vert\leq\vert B\vert## then there exist an injection from ##A## to ##B##?
Terrell said:why is not always true that if ##\vert A\vert\leq\vert B\vert## then there exist an injection from ##A## to ##B##?
I think saying that me saying that it's not always true is too strong of a statement, but what really happened is I couldn't find any sources that mentions that it must be an if and only if statement. Thank you a lot! It has cause me a lot of unnecessary thinking lol!Math_QED said:Who told you that?
By definition ##|A| \leq |B|## iff there exists an injection ##A \to B##
Terrell said:I think saying that me saying that it's not always true is too strong of a statement, but what really happened is I couldn't find any sources that mentions that it must be an if and only if statement. Thank you a lot! It has cause me a lot of unnecessary thinking lol!
Cardinality refers to the number of elements in a set. It is important in mathematics and computer science because it helps us understand the size and structure of a set, and can be used to compare and classify sets.
A one-to-one mapping, also known as a bijection, is a function where each element in the domain is paired with exactly one element in the range. An onto mapping, also known as a surjection, is a function where every element in the range has at least one corresponding element in the domain. In other words, a one-to-one mapping is a perfect pairing, while an onto mapping is a complete pairing.
If there exists a one-to-one correspondence between the elements of two sets, then they have the same cardinality. This means that each element in one set is paired with a unique element in the other set, and vice versa.
Yes, it is possible for a set to have a higher cardinality than its power set. For example, the set of real numbers has a higher cardinality than its power set, which is the set of all possible subsets of real numbers.
The Cantor-Bernstein-Schroder theorem states that if there exists a one-to-one mapping between two sets, then those sets have the same cardinality. This theorem helps us compare the sizes of sets and is often used in proofs involving cardinality.