Prove that if A and B are any countably infinite set, then is countably infinite.
If are any countably infinite sets, then is countably infinite.
are any countably infinite sets.
A function is said to be one-to-one function if the distinct elements in domain must be mapped with distinct elements in co-domain.
A function is onto function if each element in co-domain is mapped with atleast one element in domain.
Let us consider set as any countably infinite sets.
Since, is countably infinite set the elements of the set are listed as shown below.
Similarly, the elements of the set are listed as shown below
Now, let us consider the cartesian product of the two set .
The elements of are represented in a grid form as shown below.
Now, define a function from by starting to count at and moving horizontally, diagonally and vertically as shown below
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