# Discrete Mathematics: An Active Approach to Mathematical Reasoning

## AppendixANotation

Symbol Description Location
$$x\in S$$ $$x$$ is an element of $$S$$ Paragraph
$$x\notin S$$ $$x$$ is not an element of $$S$$ Paragraph
$$\{x\in S : P(x)\}$$ $$x$$ in $$S$$ such that $$x$$ has property $$P$$ Paragraph
$$A\subseteq S$$ $$A$$ is a subset of $$S$$ Definition 1.2.2
$$A\nsubseteq S$$ $$A$$ is not a subset of $$S$$ Paragraph
$$\mathbb{R}$$ the set of real numbers Item
$$\mathbb{Z}$$ the set of integers Item
$$\mathbb{Q}$$ the set of rational numbers Item
$$\mathbb{N}$$ the set of natural numbers Item
$$\mathbb{Z}^+$$ the set of positive integers Item
$$\mathbb{Z}^{nonneg}$$ the set of nonnegative integers Item
$$\mathbb{R}^+$$ the set of positive real numbers Item
$$\mathbb{R}^{nonneg}$$ the set of nonnegative real numbers Item
$$A\times B$$ the product of $$A$$ and $$B\text{;}$$ $$\{(a, b) : a\in A, b\in B\}$$ Definition 1.2.6
$$aRb$$ $$a$$ is related to $$b$$ Paragraph
$$\sim p$$ not $$p$$ Item
$$p\wedge q$$ $$p$$ and $$q$$ Item
$$p\vee q$$ $$p$$ or $$q$$ Item
$$\mathbf{t}$$ a statement that is always true; tautology Paragraph
$$\mathbf{c}$$ a statement that is always false; contradiction Paragraph
$$P\equiv Q$$ $$P$$ is logically equivalent to $$Q$$ Definition 2.1.10
$$p\rightarrow q$$ if $$p$$ then $$q$$ Item
$$\therefore$$ therefore Assemblage
$$\forall$$ for all; universal quantifier Item
$$\exists$$ there exists; existential quantifier Item
$$\mathbb{Q}$$ the set of ratioanl numbers Definition 4.2.1
$$\mathbb{R}\setminus\mathbb{Q}$$ the set of irratioanl numbers Definition 4.2.2
$$d\mid n$$ $$d$$ divides $$n$$ Paragraph
$$d$$ does not divide $$n$$ Paragraph
$$n \text{ div } d$$ quotient when $$n$$ is divided by $$d$$ Paragraph
$$n \text{ mod } d$$ remainder when $$n$$ is divided by $$d$$ Paragraph
$$\sum_{k=1}^{n}a_k$$ the sum of $$a_k$$ from $$k=1$$ to $$n$$ Assemblage
$$\prod_{k=1}^{n}a_k$$ Paragraph
$$\binom{n}{r}$$ $$n$$ choose $$r$$ Definition 5.1.6
$$A\cup B$$ $$A$$ union $$B$$ Definition 6.1.4
$$A\cap B$$ $$A$$ intersect $$B$$ Definition 6.1.6
$$A-B$$ $$A$$ minus $$B\text{;}$$ the difference of set $$A$$ and $$B$$ Definition 6.1.8
$$A^C$$ the complement of $$A$$ Definition 6.1.10
$$\bigcup_{i=1}^{n}A_i$$ the union $$A_1\cup A_2\cup\cdots \cup A_n$$ Paragraph
$$\bigcap_{i=1}^{n}A_i$$ the intersection $$A_1\cap A_2\cap\cdots \cap A_n$$ Paragraph
$$\mathcal{P}(A)$$ the power set of $$A$$ Definition 6.1.13
$$\Leftrigharrow$$ if and only if in proofs Paragraph
$$|S|$$ the number of elements in $$S$$ Paragraph
$$\text{Im}(f)$$ the image of $$f$$ Definition 7.1.6
$$f^{-1}(x)$$ the inverse of function $$f$$ Theorem 7.2.13
$$x R y$$ $$x$$ is related to $$y$$ Paragraph
$$m\equiv n \mod d$$ $$m$$ is congruent to $$n$$ mod $$d\text{;}$$ $$d\mid (m-n)$$ Paragraph
$$[a]$$ the equivalence class of $$a$$ Paragraph
$$P(n, r)$$ the number of $$r$$-permutations from a set of $$n$$ elements Definition 9.2.8
$$\binom{n}{r}$$ $$n$$ choose $$r$$ Definition 9.5.1