One-to-one correspondence

[rethipher]

What is a one-to-one correspondence or one to one mapping? I have heard the later term used plenty of times in linear algebra classes I've taken, i.e. there is a one to one mapping from a subspace to another. But I've never really understood what that meant entirely. Are the two above phrasees the same, or different? And if they are different how are they different? Quick sidenote: this is not homework of any kind, no problems/grades or any such thing. I do self study in my down time when I'm not in school, and this came up in a math book I'm looking at and I think I need to fully understand what it means so I get the full understanding and not just a superficial understanding that I can confuse for real understanding. Thanks for your time and any answer!

 

[Number Nine]

The wiki article on Bijection should cover it pretty thoroughly. The premise is as follows: Given two sets (collections of objects), a one-to-one correspondence (bijection) describes a construction where every element in one set is associated with one and only one element of the other set, and vice-versa. It's difficult to explain properly without the concept of a function, but I'm not sure how much set theory you've been exposed to.

 

[jedishrfu]

if you think of a space as a collection of points then a one to one mapping from one space to another is a means of associating a pt in one space with a pt in another for all pts and vice versa.

A simple set example would be to associate the letters of the alphabet with the range of integers from 1 to 26. there is no letter without a corresponding number and there is no number without a corresponding letter.

wikipedia describes it in more detail:

http://en.wikipedia.org/wiki/One-to-one_correspondence

 

[SteveL27]

It's worth noting that 1-1 correspondence is not the same as 1-1 mapping.

In a 1-1 mapping, different elements of the domain go to different elements of the range.

A 1-1 correspondence is a 1-1 mapping in which every element of the range gets hit by some element of the domain.

This is a confusing bit of terminology, which is why it's better to use the terms injection and bijection. An injection is what I just defined as a 1-1 mapping. A bijection is a 1-1 correspondence.

 

[CellCoree]

One-to-one correspondence, also known as one-to-one mapping, is a concept in mathematics that refers to a relationship between two sets where each element in one set is paired with exactly one element in the other set. This means that there is a unique and distinct connection between the elements of the two sets.

In linear algebra, one-to-one mapping refers to a specific type of function or transformation that preserves the one-to-one correspondence between the elements of two vector spaces. This means that each element in the first vector space is mapped to exactly one element in the second vector space.

The two phrases, one-to-one correspondence and one-to-one mapping, are often used interchangeably and refer to the same concept. They both describe a one-to-one relationship between elements in two sets or vector spaces.

It is important to fully understand the concept of one-to-one correspondence in order to have a strong understanding of mathematical concepts such as functions, transformations, and mappings. It is also a fundamental concept in many areas of mathematics, including algebra, geometry, and analysis.

I hope this explanation helps clarify the concept of one-to-one correspondence for you. If you have any further questions, please do not hesitate to ask.