(a) (b) Let X == {5, {5,6}}. How many elements does the set X have? Give the power set P(X). Prove that the sets x-5 A = {x Є Z: 6 € Z} and B = {3k: k Є N} are disjoint, i.e. AnB = 0. (c) (d) (e) Prove by induction that for all natural N, N Σ k (k + 1) = ½ N(N + 1)(N +2). k=1 Indicate clearly where you use the inductive hypothesis. Use proof by contradiction to show that 2 is irrational. Write the number 6.29474747474... as an infinite sum. Then use an appropriate formula to write it as a quotient of two integers.

Write out the addition and multiplication tables for 5.

Prove by induction that 1+2n3n for n1.

49. a. The binomial coefficients are defined in Exercise of Section. Use induction on to prove that if is a prime integer, then is a factor of for . (From Exercise of Section, it is known that is an integer.) b. Use induction on to prove that if is a prime integer, then is a factor of .