Show that set of positive integersand set of all integers have same cardinality.

To show that the set of positive integers and the set of all integers have the same cardinality, we need to establish a one-to-one correspondence (or bijection) between the two sets.

Step 1: Define the two sets Let's denote the set of positive integers as P = {1, 2, 3, 4, ...} and the set of all integers as I = {..., -3, -2, -1, 0, 1, 2, 3, ...}.

Step 2: Define a function that maps one set to the other We can define a function f: P -> I that maps each positive integer to an integer in such a way that every integer is paired with exactly one positive integer.

One possible function is f(n) = 2n for n odd and f(n) = -n for n even. This function pairs each positive integer with an integer as follows:

f(1) = 2, f(2) = -2, f(3) = 6, f(4) = -4, f(5) = 10, f(6) = -6, ...

Step 3: Show that the function is a bijection This function is a bijection because it pairs each positive integer with exactly one integer (so it's a function), and each integer is paired with exactly one positive integer (so it's a one-to-one correspondence).

Therefore, the set of positive integers and the set of all integers have the same cardinality.