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## Chapter 1. Preliminaries ### 1.1 Sets A **set** is a well-defined collection of objects; these objects are called **elements** or **members** of the set. If the element $x$ is a member of the set $A$, we write $x \in A$ If $x$ is not a member of $A$, we write $x \notin A$ If a set has a fi...

## Chapter 1. Preliminaries ### 1.1 Sets A **set** is a well-defined collection of objects; these objects are called **elements** or **members** of the set. If the element $x$ is a member of the set $A$, we write $x \in A$ If $x$ is not a member of $A$, we write $x \notin A$ If a set has a finite number of elements, the number of elements is denoted by $|A|$. A set can be specified by listing its elements (separated by commas) between braces. Thus the set $A$ consisting of the numbers 4, 5, and 6 is written as $A = \{4, 5, 6 \}$ The set of all positive integers is denoted by $\mathbb{N}$: $\mathbb{N} = \{1, 2, 3,...\}$ The set of all integers is denoted by $\mathbb{Z}$: $\mathbb{Z} = \{..., -2, -1, 0, 1, 2,...\}$ The set of all real numbers is denoted by $\mathbb{R}$. **Example 1.** Let $A = \{x \in \mathbb{R} : 0 \le x \le 1 \}$ That is, $A$ is the set of real numbers $x$ such that $0 \le x \le 1$. Let $B = \{x \in \mathbb{R} : x^2 - 3x + 2 = 0\}$ Then, $B = \{1, 2 \}$. We say that $A$ is a **subset** of $B$, written $A \subseteq B$, if every element of $A$ is an element of $B$. When $A \subseteq B$ and $A \ne B$, we say that $A$ is a **proper subset** of $B$, written $A \subset B$. **Example 2.** Let $A = \{1, 3, 5\}$ and $B = \{5, 4, 3, 1\}$. Then $A \subseteq B$, since every element of $A$ is also an element of $B$. The set that contains no elements is called the **empty set**, and is denoted by $\phi$. If $A$ is any set, then $\phi \subseteq A$. Two sets are **equal** if they contain the same elements. If $A$ and $B$ are sets, then $A = B$ if and only if $A \subseteq B$ and $B \subseteq A$. If $A$ and $B$ are sets, then the **union** of $A$ and $B$, written $A \cup B$, is the set of all elements that belong to $A$ or $B$; that is, $A \cup B = \{x : x \in A \text{ or } x \in B\}$ The **intersection** of $A$ and $B$, written $A \cap B$, is the set of all elements that belong to both $A$ and $B$; that is, $A \cap B = \{x : x \in A \text{ and } x \in B \}$ The **difference** of $A$ and $B$, written $A - B$, is the set of all elements that belong to $A$ but not to $B$; that is $A - B = \{x : x \in A \text{ and } x \notin B \}$ **Example 3.** Let $A = \{1, 2, 3\}$ and $B = \{2, 3, 4\}$. Then, $A \cup B = \{1, 2, 3, 4 \}$ $A \cap B = \{2, 3 \}$ $A - B = \{1 \}$ $B - A = \{4 \}$ **Definition.** Let $A$ be a set. The **power set** of $A$, written $P(A)$, is the set of all subsets of $A$. **Example 4.** Let $A = \{a, b, c \}$. Then the power set of $A$ is $P(A) = \{ \phi, \{a \}, \{b \}, \{c \}, \{a,b \}, \{a,c \}, \{b,c \}, \{a,b,c \} \}$ ### 1.2 Functions **Definition.** A **function** from a set $A$ to a set $B$ is a rule that assigns to each element $x$ in $A$ exactly one element $y$ in $B$; the element $y$ is denoted by $f(x)$. We write $f: A \rightarrow B$ to indicate that $f$ is a function from $A$ to $B$. The set $A$ is called the **domain** of $f$, and the set $B$ is called the **codomain** of $f$. The **range** of $f$ is the set $\{ f(x) : x \in A \}$ Note that the range of $f$ is a subset of the codomain of $f$. If $f: A \rightarrow B$ and $g: B \rightarrow C$, we can define a new function $g \circ f: A \rightarrow C$, called the **composition** of $g$ and $f$, defined by $(g \circ f)(x) = g(f(x))$ for all $x \in A$. **Definition.** A function $f: A \rightarrow B$ is **one-to-one** (or **injective**) if $f(x_1) = f(x_2)$ implies that $x_1 = x_2$. **Definition.** A function $f: A \rightarrow B$ is **onto** (or **surjective**) if for every $y \in B$, there exists an element $x \in A$ such that $f(x) = y$. **Definition.** A function $f: A \rightarrow B$ is a **one-to-one correspondence** (or **bijective**) if $f$ is one-to-one and onto.