Math Problems Vault

Problem 1.2.1

Let $\left\{ A_{i}: i \in I\right\}$ be a collection of sets. Prove 'De Morgan's Laws' :

\left(\bigcup _{i} A_{i}\right)^{\complement } = \bigcap _{i} A_{i}^{\complement }, \quad \left(\bigcap _{i} A_{i}\right)^{\complement } = \bigcup _{i} A_{i}^{\complement }

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Problem 1.2.2

Let $A$ and $B$ belong to some $\sigma$ -field $\mathcal{F}$ . Show that $\mathcal{F}$ contains the sets $A \cap B, A \backslash B$ , and $A \triangle B$ .

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Problem 1.2.3

A conventional knock-out tournament (such as that at Wimbledon) begins with $2^{n}$ competitors and has $n$ rounds. There are no play-offs for the positions $2,3, \ldots , 2^{n}-1$ , and the initial table of draws is specified. Give a concise description of the sample space of all possible outcomes.

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Problem 1.2.4

Let $\mathcal{F}$ be a $\sigma$ -field of subsets of $\Omega$ and suppose that $B \in \mathcal{F}$ . Show that $\mathcal{G} = \left\{ A \cap B: A \in \mathcal{F}\right\}$ is a $\sigma$ -field of subsets of $B$ .

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Problem 1.2.5

(a)(b)(c)(d)

Which of the following are identically true? For those that are not, say when they are true.

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(a)

$A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$ ;

(b)

$A \cap (B \cap C) = (A \cap B) \cap C$ ;

(c)

$(A \cup B) \cap C = A \cup (B \cap C)$ ;

(d)

$A \backslash (B \cap C) = (A \backslash B) \cup (A \backslash C)$ .