Discrete Mathematic

Problem 1 ( 2 pts): Prove that the following two sets A and B are equinumerous: (a) A={nN:n is divisible by 3} and B={nN:n is divisible by 4} (b) A={nN:n is not divisible by 3} and B={nN:n is not divisible by 4} Problem 2 (1 pt): Prove that these two sets A, and B of functions are equinumerous, where A consists of all functions f:NN, and B consists of all functions g:NN that are monotone (that is, for every m,nN, if mn then g(m)g(n)) Problem 3(1pt) : Prove that the set of functions g:NN that are monotone is not countable. Problem 4 (1pt): Prove that the following set A is countable: A consists of all infinite sequences a0,a1, that are monotone and such that for every i=0,1,,ai{0,1,2}. Problem 5 (3 pts): Find the limits of the following sequences (show your work, explain your reasoning; you may use properties discussed in class) (a) limnn+13n/2 (b) limnn+1log5(3n2+2) (c) limnn+1n22n+4 (d) limnn2+1n22n+4 (f) limnn+1n+log3(n+1) Problem 6(1pt) : Use mathematical induction to prove that n2n! holds for every n4