# Introduction to Proofs: An Active Exploration of Mathematical Language

## AppendixANotation

Symbol Description Location
$$\therefore$$ therefore Assemblage
$$\neg p$$ not $$p\text{;}$$ negation Item
$$p\wedge q$$ $$p$$ and $$q\text{;}$$ conjunction Item
$$p\vee q$$ $$p$$ or $$q\text{;}$$ disjunction (inclusive) Item
$$p\veebar q$$ $$p$$ exclusive or $$q\text{;}$$ disjunction (exclusive) Item
$$p\rightarrow q$$ if $$p$$ then $$q\text{;}$$ conditional Item
$$p\leftrightarrow q$$ $$p$$ if and only if $$q\text{;}$$ biconditional Item
$$p // q$$ $$p$$ nand $$q\text{;}$$ not both Item
$$\mathbb{t}$$ tautology; always true Paragraph
$$\mathbb{c}$$ contradiction; always false Paragraph
$$P\equiv Q$$ $$P$$ is logically equivalent to $$Q$$ Definition 1.3.14
$$x\in S$$ $$x$$ is an element of set $$S$$ Paragraph
$$x\notin S$$ $$x$$ is not an element of set $$S$$ Paragraph
$$B\subseteq A$$ $$B$$ is a subset of $$A$$ Paragraph
$$B\nsubseteq A$$ $$B$$ is not a subset of $$A$$ 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
$$\emptyset$$ empty set; set with no elements Paragraph
$$\forall$$ for all; universal quantifier Item
$$\exists$$ there exists; existential quantifier Item
$$\mathbb{R}\setminus\mathbb{Q}$$ the set of irrational numbers Definition 3.2.2
$$d\mid n$$ $$d$$ divides $$n$$ Paragraph
$$\sum_{k=1}^{n}a_k$$ the sum $$a_1+a_2+\cdots +a_n$$ Assemblage
$$\prod_{k=1}^{n}a_k$$ the product $$a_1\cdot a_2\cdot a_3\cdots a_n$$ Paragraph
$$n!$$ $$n$$ factorial; $$(n)(n-1)\cdots(2)(1)$$ Paragraph
$$A\cup B$$ the union of $$A$$ and $$B$$ Paragraph
$$A\cap B$$ the intersection of $$A$$ and $$B$$ Paragraph
$$A-B$$ the set difference $$A$$ minus $$B$$ Paragraph
$$A\setminus B$$ the set difference $$A$$ minus $$B$$ Paragraph
$$A^C$$ the complement of $$A$$ Paragraph
$$\mathcal{P}(A)$$ The power set of $$A$$ Definition 5.2.9
$$A\times B$$ the Cartesian product of $$A$$ and $$B$$ Paragraph
$$x R y$$ $$x$$ is related to $$y$$ Paragraph
$$m\equiv n \mod d$$ $$m$$ is congruent to $$n$$ mod $$d$$ Definition 5.4.11
$$[a]$$ the equivalence class of $$a$$ Paragraph
$$\binom{n}{r}$$ $$n$$ choose $$r\text{;}$$ binomial coefficient Definition 6.1.1
$$|A|$$ cardinality of set $$A$$ Paragraph