Algebra/Chapter 15/Exercises

A set of exercises related to concepts from Chapter 15.

This set contains 18 exercises (including the Conceptual Questions)

Exercises

Section 15.1

(★) 15.1 (Discrete Function) Consider the function $f:\{1,2,3,4,5\}\rightarrow \{1,2,3,4,5\}$, given by

$$f={\begin{bmatrix}1&2&3&4&5\\1&5&3&2&3\end{bmatrix}}$$

1. Evaluate the following:
i. $f(2)$
ii. $f(5)$
iii. $f(6)$
iv. $f(4.5)$
2. Find an n in the domain of f such that f(n) = 5.
3. Find an n in the domain of f such that f(n) = n.
4. Find an element in the codomain of f that is not in its range.

(★) 15.2 (Injective and Surjective I) All of the functions below have the domain and codomain of $\{1,2,3,4,5\}$. Determine if each of them are only injective, only surjective, bijective, or neither injective or surjective.

(★) 15.3 (Injective and Surjective II) All of the functions below are determined by $f:\{1,2,3,4,5\}\rightarrow \{1,2,3\}$. Determine if each of them are only injective, only surjective, bijective, or neither injective or surjective.

(★) 15.4 (Injective and Surjective III) All of the functions below are determined by $f:\{1,2,3,4\}\rightarrow \{1,2,3,4,5\}$. Determine if each of them are only injective, only surjective, bijective, or neither injective or surjective.

(★) 15.5 (Injective and Surjective IV) Write out all of the functions determined by $f:\{1,2,3,4\}\rightarrow \{a,b\}$.
1. How many functions are possible?
2. How many of them are only injective, only surjective, bijective, and neither respectively?

(★) 15.6 (Injective and Surjective V) Write out all of the functions determined by $f:\{1,2\}\rightarrow \{a,b,c,d\}$.
1. How many functions are possible?
2. How many of them are only injective, only surjective, bijective, and neither respectively?

Section 15.2

(★) 15.7 (Multiples of 6) Use induction to prove that $7^{n}-1$ is a multiple of 6 for all natural numbers n.

(★) 15.8 (Sum of Odd Numbers) Use induction to prove that $1+3+5+...(2n-1)=n^{2}$.

(★) 15.9 (Sum of Squares) Use induction to prove that $1^{2}+2^{2}+3^{2}+...n^{2}={\frac {n(n+1)(2n+1)}{6}}$.

(★★) 15.10 (2 to the n) Use induction to prove that $2^{n+1}>n^{2}$ for all positive integers.

(★★) 15.11 (Bernoulli's Inequality) Bernoulli's Inequality approximates powers of $(x+1)$. This inequality is of the form:

$(1+x)^{k}\geq 1+kx$, where k and x are integers

.

Use induction to prove this inequality.

Section 15.3

(★) 15.12 (Recursive Function) Consider the function $f:\mathbb {N} \rightarrow \mathbb {N}$ given by $f(0)=1$ and $f(n+1)=3*f(n)$. What is $f(14)$?

(★★) 15.13 (Functional Equation) A function $f(x)$ defined on the positive integers satisfies $f(1)=2000$ and $f(1)+f(2)+f(3)+f(4)+...f(n)=n^{2}f(n)$. Calculate $f(2000)$.

Section 15.4

(★) 15.14 (Sigma/Pi Notation) Write the following sequences in either sigma or pi notation.

(★) 15.15 (Expanding Sigma/Pi Notation) Expand the following sums and products.

1. $\prod _{i=1}^{9}i$
2. $\prod _{i=1}^{3}(i+x)$
3. $6!$

(★★) 15.16 (The Gamma Function) The Gamma Function is of the form $\Gamma (x)=(x-1)!$. It is also has the following properties:

• $\Gamma (x+1)=x\Gamma (x)$
• $\Gamma ({\frac {1}{2}})={\sqrt {\pi }}$

Use this information to find the following.

1. $\Gamma (5)$
2. $\Gamma ({\frac {3}{2}})$
3. $({\frac {1}{2}})!$
4. $({\frac {5}{2}})!$
5. $\Gamma ({\frac {n}{2}})$, where $n\in \mathbb {N}$

(★★) *15.17 (n Factorial) Prove that $n!<n^{n}$ when $n>1$.

Section 15.8

15.18 (Twelve Days of Christmas) In the song The Twelve Days of Christmas, my true love gave me 1 gift on the first day, then 2 gifts and 1 gift on the second day, then 3 gifts, 2 gifts, and 1 gift on the third day, and so forth. How many gifts in total did my true love give to me on all 12 days?