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Waiting 'Til the Midnight Hour: A Narrative History of Black Power in America, by Peniel E. What an opportunity for cross-curriculum collaboration and providing a simple real world scenario! Roosevelt stated that freedom from want "translated into world terms, means economic understandings which will secure to every nation a healthy peacetime life for its inhabitants-everywhere in the world". Then \begin{enumerate} \item The union of $\mathbf{X}$ is the set $\bigcup \mathbf{X} = \{\, z \mid \exists a \in A \colon z \in X_a \,\}$.\index{union} \item The intersection of $\mathbf{X}$ is the set $\bigcap \mathbf{X} = \{\, z \mid \forall a \in A \colon z \in X_a \,\}$.\index{intersection} \item The cross product of $\mathbf{X}$ is the set of sequences (indexed by $A$) $\prod \mathbf{X} = \prod_{a \in A} X_a = \{\, (\, z_a \mid a \in A \,) \mid \forall a \in A \colon z_a \in X_a \,\}$.\index{cross product} \index{$\prod$} \end{enumerate} We will denote the cross product of a set $X$ with itself taken $n$ times ({\em i.e.\/} the set of all sequences of length $n$ of elements of $X$) by $X^n$. \index{$X^n$} \end{defn} \begin{defn} If $X$ is any set, a {\em $k$-place relation on $X$\/} is a subset $R \subseteq X^k$. \index{relation, $k$-place} \end{defn} For example, the set $E = \{\, 0, 2, 3, \dots \,\}$ of even natural numbers is a $1$-place relation on $\mathbb{N}$, $D = \{\, (x,y) \in \mathbb{N}^2 \mid x \text{\ divides\ } y \,\}$ is a $2$-place relation on $\mathbb{N}$, and $S = \{\, (a,b,c) \in \mathbb{N}^3 \mid a + b = c \,\}$ is a $3$-place relation on $\mathbb{N}$. $2$-place relations are usually called binary relations.\index{relation, binary} \begin{defn} A set $X$ is {\em finite\/}\index{finite} if there is some $n \in \mathbb{N}$ such that $X$ has $n$ elements, and is {\em infinite\/}\index{infinite} otherwise. $X$ is {\em countable\/}\index{countable} if it is infinite and there is a 1-1 onto function $f: \mathbb{N} \to X$, and {\em uncountable\/}\index{uncountable} if it is infinite but not countable. \end{defn} Various infinite sets occur frequently in mathematics, such as $\mathbb{N}$\index{$\mathbb{N}$} (the natural numbers), $\mathbb{Q}$\index{$\mathbb{Q}$} (the rational numbers), and $\mathbb{R}$\index{$\mathbb{R}$} (the real numbers).
Pages: 256
Publisher: Hamish Hamilton; Reprint edition (March 1986)
ISBN: 0241115930
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