Given a positive integer n, use Exercise 1 to find π(n).Exercise 1Use the principle of inc...

Given a positive integer n, use Exercise 1 to find π(n).

Exercise 1

Use the principle of inclusion-exclusion (Exercise 2 of Appendix B) to show that

where p1, p2, …., pr are the primes less than or equal to .(Hint: Let property Pi be the property that an integer is divisible by pi.)

Exercise 2

In this exercise, we develop the principle of inclusion-exclusion. Suppose that S is a set with nelements and let P1, P2,…, P1 be t different properties that an element of S may have. Show that the number of elements of S possessing none of the t properties is

where  is the number of elements of S possessing all of the properties . The first expression in brackets contains a term for each property, the second expression in brackets contains terms for all combinations of two properties, the third expression contains terms for all combinations of three properties, and so forth, (Hint: For each element of S, determine the number of times it is counted in the above expression. If an element has k of the properties, show that it is counted  times; this is 0 when k > 0. by Exercise 3.)

Exercise 3

Find all primes that are the difference of the fourth powers of two integers