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Real Analysis: Math 4050-4060

Valerio De Angelis

Section 1.6 Exercises

Exercise 1.6.1.

Decide if the statements are true or false.
  1. 5 is odd and \(3\leq 3\text{.}\)
  2. 5 is odd or \(3 \lt 3\text{.}\)
  3. If 5 is prime, then 9 is odd.
  4. If 9 is prime, then 5 is odd.
  5. If 5 is prime, then 9 is even.
  6. If 9 is prime, then 5 is even.
  7. If 3 is even or 11 is prime, then \(5\leq 6\text{.}\)
  8. If 3 is even and 11 is prime, then \(5\leq 6\text{.}\)

Exercise 1.6.2.

Write the negation of each statement using plain English.
  1. Every book has at least 10 pages.
  2. There are books with more than 10 pages.
  3. Some pencils cannot be used underwater.
  4. Some pencils can be used underwater.
  5. Everybody likes chocolate.
  6. Some people like chocolate.

Exercise 1.6.3.

Let \(B\) be the set of all books, and for \(b\in B\text{,}\) let \(n(b)\) be the number of pages of \(b\text{.}\) Use the universal quantifiers \(\exists\) and \(\forall\text{,}\) the implication symbol \(\Rightarrow\text{,}\) and the "such that" symbol \(\ni\) to write both the statements and their negation in Exercise 1.6.2 a. and b. without using any English words.

Exercise 1.6.4.

Let \(P\) be the set of all people, and \(Q\) the set of people who like chocolate. Use the symbols of set theory to write both the statements and their negation in Exercise 1.6.2 e. and f. without using any English words.

Exercise 1.6.5.

We will prove that all horses are of the same color. We proceed by induction on the number of horses that exist in the world. If that number is 1, then of course there is only one possible color for that single horse. Suppose now that in every set of \(n\) horses, each horse has the same color, and consider a set of \(n+1\) horses. We can number the horses from \(1\) to \(n+1\text{,}\) \(H_1,H_2, \ldots, H_n, H_{n+1}\text{.}\) Consider the two sets of \(n\) horses each \(\{H_1,H_2, \ldots, H_n\}\) and \(\{H_2, H_3, \ldots, H_{n+1}\}\text{.}\) By induction, all the horses in the first set have the same color as \(H_2\) and all the horses in the second set have the same color as \(H_2\text{.}\) Therefore all horses in both sets have the same color as \(H_2\text{,}\) and we are done. We have proved that all horses in the world have the same color. Are you convinced? If not, what is wrong with the above proof?

Exercise 1.6.6.

Prove that \((A\setminus B)^c=A^c\cup B\)

Exercise 1.6.9.

Let \(A,B\) be sets and \(f:A\longrightarrow B\text{.}\) Suppose \(X\) and \(Y\) are subsets of \(A\text{.}\)
  1. How are the subsets \(f(X\cup Y)\) and \(f(X)\cup f(Y)\) related?
  2. How are the subsets \(f(X\cap Y)\) and \(f(X)\cap f(Y)\) related?

Exercise 1.6.10.

Let \(A,B\) be sets and \(f:A\longrightarrow B\text{.}\) Suppose \(U\) and \(V\) are subsets of \(B\text{.}\)
  1. How are the subsets \(f^{-1}(U\cup V)\) and \(f^{-1}(U)\cup f^{-1}(V)\) related?
  2. How are the subsets \(f^{-1}(U\cap V)\) and \(f^{-1}(U)\cap f^{-1}(V)\) related?

Exercise 1.6.11.

Let \(A\) and \(B\) be sets, and \(f:A\rightarrow B\text{.}\)
  1. Suppose that \(Y\) is a subset of \(B\text{.}\) Prove that \(f^{-1}\left(Y^c\right)=\left(f^{-1}(Y)\right)^c .\)
  2. Suppose now that \(X\) is a subset of \(A\text{.}\) Prove that:
    1. If \(f\) is injective, then \(f(X^c)\subseteq \left(f(X)\right)^c\)
    2. If \(f\) is surjective, then \(\left(f(X)\right)^c\subseteq f(X^c)\)
  3. What can you conclude from i. and ii. in case \(f\) is bijective?
  4. Find an example to show that if \(f\) is not injective, b.i. is false.
  5. Find an example to show that if \(f\) is not surjective, b.ii. is false.

Exercise 1.6.12.

Let \(A,B\) be sets and \(f:A\longrightarrow B\text{.}\)
  1. Prove that \(f\) has a right inverse if and only if it is surjective.
  2. Prove that \(f\) has a left inverse if and only if it is injective.

Exercise 1.6.13.

Let \(n\) be an integer. Prove that \(n(n+1)(n+2)\) is divisible by \(6\text{.}\)

Exercise 1.6.14.

  1. Compute \(\frac{7^n-4^n}{3}\) for \(n=1,2,3\) and 4
  2. Prove that for all positive integers \(n\text{,}\) \(7^n-4^n\) is divisible by 3.

Exercise 1.6.15.

Suppose \(x\in {\mathbb Q}\) and \(x\gt -1\text{.}\) Prove that \((1+x)^n\geq 1+nx\) is true for all \(n\in {\mathbb N}\) (this is called Bernoulli’s inequality).

Exercise 1.6.16.

Find an example to show that Bernoulli’s inequality is false if we drop the assumption that \(x\gt -1\text{.}\)

Exercise 1.6.17.

Let \(f(n)\) be the sum of the first \(n\) odd numbers. So \(f(1)=1\text{,}\) \(f(2)=1+3=4\text{,}\) \(f(3)=1+3+5=9\text{,}\) and so on.
  1. Find a formula for \(f(n)\) as a function of \(n\text{.}\)
  2. Prove the formula by induction.

Exercise 1.6.18.

Prove that for all positive integers \(n\text{,}\)
\begin{equation*} \sum_{k=1}^n k^2=\frac{n(n+1)(2n+1)}{6} \end{equation*}

Exercise 1.6.19.

Prove that the Principle of Mathematical Induction is equivalent to the Well-Ordering Principle.
To prove that \(\mbox{WOP}\Rightarrow \mbox{PMI}\text{,}\) start assuming that WOP is true. Then assume that PMI is false, which means that we can find some sequence of statements \({\cal P}(n)\) for which a. and b. of Section Subsection 1.5.4 are true, but c. is false. Now construct the set \(S=\{n| {\cal P}(n) \mbox{ is false }\}\text{.}\)
To prove that \(\mbox{PMI}\Rightarrow \mbox{WOP}\text{,}\) start assuming that PMI is true. Suppose there is a set \(S\) of positive integers without a least element, and consider the statement \({\cal P}(n): i \not \in S \mbox{ for } 1\leq i \leq n\text{.}\) Use the PMI to show that \(S\) must be empty.
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