Using that result, prove by induction that for any positive integer ?, an infinite ?-dimensional array is countable.

Problem 12 Use Mathematical Induction to prove that 6 n³-n whenever n is a positive integer.

1. Prove that for all integers n, n2-n+3 is odd. (Write a standalone self-contained proof using definitions, properties, and axioms. Do not use mathematical induction and do not usc any lemma.)

Suppose that in a particular Finite course, 4/7 of students plan on majoring in math, 25/42 plan on majoring in communications, and 2/7 plan on double majoring in math and communications. How many want to major in exactly one of communications or math?

10) Identify the mistake in the following “proof" by mathematical induction. (Fake) Theorem: For any positive integer n, all the numbers in a set of n numbers are equal to each other. Proof (by mathematical induction): It is obviously true that all the numbers in a set consisting of just one number are equal to each other, so the basis step is true. For the induction step, let A = {a1, a2, a3, . .., an41} be any set of n +1 numbers. Form two subsets each of size n: B = {a1,a2, A3, . numbers in A except an+1, and C consists of all the numbers in A except a2.) By the induction hypothesis, all the numbers in B equal a1 and all the numbers in C equal a1, since both sets have only n numbers. But every number in A is in B or C, so all the numbers in A equal a1; thus all are equal to each other. .., a,} and C = {a1, az, a4, . .. , an+1}. (Precisely, B consists of all the

First, consider the following infinite collection of real numbers. Describe in your own words howthese numbers are constructed (that is, describe the procedure for generating this list of numbers). Then,using Cantor s diagonalization argument, find a number not on the list. Justify your answer.0.123456789101112131415161718 . . .0.2468101214161820222426283032 . . .0.369121518212427303336394245 . . .0.4812162024283236404448525660 . . .0.510152025303540455055606570 . .