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

Can you conclude that A=B if A and B are two sets with the same power set?

An employee joined a company in 2009 with a starting salary of \( 50,000 . Every year this employee receives a raise \)1000 of plus 5% of the salary of the previous year.

1. Set up a recurrence relation for the salary of this employee n years after 2009
2. What will the salary of this employee be in 2017?
3. Find an explicit formula for the salary of this employee n years after 2009

Determine whether each of these functions is a bijection from R to R.

1. $f\left(x\right)=-3x+4$
2. $f\left(x\right)=-3x^2+7$
3. role="math" localid="1668414131444" $f\left(x\right)=(x+1)/(\mathrm{x}+2)$
4. role="math" localid="1668414147841" $f\left(x\right)=x^5+1$

Let A be a matrix. Show that the matrix is symmetric. [Hint: Show that this matrix $\mathit{A}{\mathit{A}}^{\mathbf{t}}$ equals its transpose with the help of Exercise 17b.]

Suppose that A is a countable set. Show that the set B is also countable if there is an onto function f from A to B.

Prove the first distributive law from Table 1 by showing that if A, B, and C are set, then A∪(B ∩ C) = (A∪B) ∩ (A∪C).

Find a recurrence relation for the balance owed at the end of months on a loan $5000 of at a rate of 7% if a payment of is made each month. [Hint: Express B(k) in terms of B(k - 1) the monthly interest is (0.07 / 12) B (k - 1).]

How many elements does each of these sets have where a and b are distinct elements?

(a) \({\bf{P}}\left( {\left\{ {{\bf{a,b,}}\left\{ {{\bf{a,b}}} \right\}} \right\}} \right)\)

(b) \(P\left( {\left\{ {\phi ,a,\left\{ a \right\},\left\{ {\left\{ a \right\}} \right\}} \right\}} \right)\)

(c) \(P\left( {P\left( \phi \right)} \right)\)

Determine whether each of these functions is a bijection from R to R.

1. $f\left(x\right)=2x+1$
2. $f\left(x\right)=x^2+1$
3. $f\left(x\right)=x$
4. $f\left(x\right)=\left(x^2+1\right)/\left(x^2+2\right)$

Suppose that A is an $\mathit{n}\mathbf{\times }\mathit{n}$ matrix where n is a positive integer. Show that$\mathit{A}\mathbf{+}{\mathit{A}}^{\mathbf{t}}$ is symmetric.