One-to-one Correspondence

In mathematics, the One-to-One correspondence or bijection or the bijective function is referred to a condition when all the members of one Set say set ‘M’ is paired with the members of other set say set ‘N’. Not even a single member of any of the set remains unpaired. It means each and every member of the set is paired with the one member of other set. The one – to - one correspondence from set ‘M’ to the set ‘N’ carries the opposite function from ‘N’ to ‘M’. If the Sets ‘M’ and ‘N’ are finite in number then the presence of One-to-One Correspondence shows that they pursue the same number of element. A one to one correspondence from all Sets or all the elements of set to itself is sometimes known as Permutation. The one-to-one function plays a vital role in many areas of mathematics which include definitions of permutation group, projective map, homeomorphism, etc.
One-to-one correspondence must hold the four properties to do the exact pairing between ‘M’ and ‘N’. They can be:

1. Each member of the set ‘M’ must be matched with at least one of the element of set ‘N’.
2. None of the element of set ‘M’ should be repeated to form the pair with other element of set ‘N’.
3. Each member of set ‘N’ must be matched with at least one member of set ‘M’.

None of the element of set ‘N’ can be matched with the more than one member of set ‘M’.
If first two properties are satisfied then it will be clear that one to one correspondence makes a function with the Domain ‘M’. Functions which meet the requirement of third property are called as surjective Functions (onto N) and those Functions which are meeting the requirement of last property (which is also known as one-to-one functions) are termed as injective functions.
Or we can say one to one correspondance is a function which is both surjective and injective.