Chapter 2: Basic Structures: Sets, Functions, Sequences, Sums, and Matrices

Show that if A andBare sets|A| = |B|then P (A) = P (B) .

Specify a codomain for each of the functions in Exercise 16. Under what conditions is each of these functions with the codomain you specified onto?

A person deposits $1000 in an account that yields 9% interest compounded annually.

a) Set up a recurrence relation for the amount in the account at the end of n years.

b) Find an explicit formula for the amount in the account at the end of n years.

c) How much money will the account contain after 100 years?

Show that matrix multiplication is not commutative.

Show that if nis an integer, then n= _n/2_ + _n/2_.

(Requires calculus) Suppose that \(f\left( x \right) = {e^x}\) and \(g\left( x \right) = {e^{cx}}\) where c is a constant. Use mathematical induction together with the chain rule and the fact that \(f'\left( x \right) = {e^x}\) to prove that \({g^{\left( n \right)}} = {c^n}{e^{cx}}\) whenever \(n\) is a positive integer.

Develop an algorithm for finding a longest common subsequence of two sequence ${\mathit{a}}_{\mathbf{1}}\mathbf{,}{\mathit{a}}_{\mathbf{2}}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}{\mathit{a}}_{\mathbf{m}}$and ${\mathit{b}}_{\mathbf{1}}\mathbf{,}{\mathit{b}}_{\mathbf{2}}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}{\mathit{b}}_{\mathbf{n}}$using the values $\mathit{L}\left(i,j\right)$found by the algorithm in Exercise $\mathbf{17}$.

Show that if A,B,C and D are sets |A| = |B| and |C| = |D| then$|\mathrm{A}\times \mathrm{C}|=|\mathrm{B}\times \mathrm{D}|$

Question: Specify a codomain for each of the functions in Exercise 17. Under what conditions is each of the functions with the codomain you specified onto?

Suppose that the number of bacteria in a colony triples every hour.

a) Set up a recurrence relation for the number of bacteria after n hours have elapsed.

b) If 100 bacteria are used to begin a new colony, how many bacteria will be in the colony in 10 hours?