Analysis

Preliminary Analysis Exercise 1 Prove that $\sqrt{3}$ is irrational. Does a similar argument work to show $\sqrt{6}$ is irrational? Exercise 2 Decide which of the following represent true statements about the nature of sets. For any that are false, provide a specific example where the statement in question does not hold. If $ A_1 \supseteq A_2 \supseteq A_3 \supseteq A_4 \supseteq \cdots $ are all sets containing an infinite number of elements, then the intersection $ \bigcap_{n=1}^{\infty} A_n $ is infinite as well. If $ A_1 \supseteq A_2 \supseteq A_3 \supseteq A_4 \supseteq \cdots $ are all finite, nonempty sets of real numbers, then the intersection $ \bigcap_{n=1}^{\infty} A_n $ is finite and nonempty. $A \cap (B\cup C) = (A \cap B )\cup C$ $A \cap (B\cap C) = (A \cap B )\cap C$ $A \cap (B\cup C) = (A \cap B ) \cup (A \cap C)$ Exercise 3: De Morgan’s Law Let $ A $ and $ B $ be subsets of $ \mathbb{R} $. ...

Axiom Of Completeness Exercise 1 Let $ \mathbb{Z}_5 = {0, 1, 2, 3, 4} $ and define addition and multiplication modulo 5. In other words, compute the integer remainder when $ a + b $ and $ ab $ are divided by 5, and use this as the value for the sum and product, respectively. Show that, given any element $ z \in \mathbb{Z}_5 $, there exists an element $ y $ such that $ z + y = 0 $. The element $ y $ is called the additive inverse of $ z $. Show that, given any $ z \neq 0 $ in $ \mathbb{Z}_5 $, there exists an element $ x $ such that $ zx = 1 $. The element $ x $ is called the multiplicative inverse of $ z $. The existence of additive and multiplicative inverses is part of the definition of a field. Investigate the set $ \mathbb{Z}_4 = {0, 1, 2, 3} $ (where addition and multiplication are defined modulo 4) for the existence of additive and multiplicative inverses. Make a conjecture about the values of $ n $ for which additive inverses exist in $ \mathbb{Z}_n $, and then form another conjecture about the existence of multiplicative inverses. Exercise 2 Write a formal definition in the style of Definition 1.3.2 for the infimum or greatest lower bound of a set. Now, state and prove a version of Lemma 1.3.7 for greatest lower bounds. Exercise 3 Let $ A $ be bounded below, and define $$ B = \{ b \in \mathbb{R} : b \text{ is a lower bound for } A \}. $$ Show that $ \sup B = \inf A $. Use (1) to explain why there is no need to assert that greatest lower bounds exist as part of the Axiom of Completeness. Propose another way to use the Axiom of Completeness to prove that sets bounded below have greatest lower bounds. Exercise 4 Assume that $A$ and $B$ are nonempty, bounded above, and satisfy $B \subseteq A$. Show that $ \sup(B) \leq \sup(A) $. ...