When two sets are said to be ‘one to one’ correspondence and

Question

establish a ‘one to one’ correspondence between set of all integers and the set of all positive integers?

Let A and B be two

sets, which are in one to one correspondence, can A and B have same number of elements? Justify

your answer.

Answer 1

The set of of all integers, positive, negative or zero, is countable. In fact, we can set up the following one-to-one correspondence between and the set of all positive integers.

More, explicitly, we associate the non-negative integer with the odd number , and the negative integer with the even number ,

The symboldenotes the a one-to one correspondence.

There is a one-to-one correspondence between the set of all integers and the set of positive integers.

Let and be two sets, which are in one-to-one correspondence.

For finite sets: A one-to-one correspondence exist if and only if the sets have the same number of elements.

For infinite sets: Two infinite sets of elements have the same transfinite cardinal number if and only if there exists a one-to-one correspondence between the elements of the two sets.

Therefore, and have same number of elements.