One to One Correspondence vs One to One function

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In summary, there are two definitions of set cardinality - one based on one-to-one correspondence and the other based on one-to-one function. While they have some similarities, they are not the same and have different ways of comparing the cardinality of two sets. One-to-one function is also known as an injection, while one-to-one correspondence is known as a bijection. However, the English names for these concepts can be confusing.
  • #1
bigpunz04
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What is the difference between the two?

The topic we are currently reading about is Set Cardinality. There are a couple of definitions listed in the book that seem to define them as different properties of sets. Is there a difference between the two or are they different terms with the same meaning? See below:

Definition 1
The sets A and B have the same cardinality if and only if there is a "one-to-one correspondence" from A to B. When A and B have the same cardinality, we write |A| = |B|

Definition 2
If there is a "one-to-one function" from A to B, the cardinality of A is less than or the same as the cardinality of B and we write |A| <= |B|. Moreover, when |A|<=|B| and A and B have different cardinality, we say that the cardinality of A is less than the cardinality of B and we write |A|<|B|

Thank you!
 
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  • #2
One-to-one function is otherwise called an injection. One-to-one correspondence is called a bijection. It is an injection that is also a surjection. I agree that the English names are somewhat confusing.
 

Related to One to One Correspondence vs One to One function

What is the difference between one to one correspondence and one to one function?

One to one correspondence refers to a relationship between two sets where each element in the first set corresponds to exactly one element in the second set. This means that every element in the first set has a unique partner in the second set, but not necessarily every element in the second set has a partner in the first set. On the other hand, a one to one function is a special type of one to one correspondence where each element in the first set has a unique partner in the second set and every element in the second set has a partner in the first set.

How do you determine if a relationship is a one to one correspondence?

To determine if a relationship is a one to one correspondence, you can use a mapping diagram or a table to list out the elements of each set and check if each element in the first set has a unique partner in the second set. If there are no repeated elements in the second set, then the relationship is a one to one correspondence.

What does it mean for a function to be "one to one"?

A function is considered one to one if each element in the domain corresponds to exactly one element in the range. This means that for every input, there is only one possible output. In other words, there are no repeated elements in the range.

Can a one to one correspondence be a one to one function?

Yes, a one to one correspondence can also be a one to one function. This means that every element in the first set has a unique partner in the second set and every element in the second set has a partner in the first set.

What is the importance of one to one correspondence and one to one functions in mathematics and science?

One to one correspondence and one to one functions are important concepts in mathematics and science because they help us analyze and describe relationships between two sets of data. They also play a crucial role in many mathematical and scientific applications, such as creating accurate models, making predictions, and solving equations.

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