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Section1.2Sets and Equivalence Relations

Subsection1.2.1Set Theory

A set is a well-defined collection of objects; that is, it is defined in such a manner that we can determine for any given object \(x\) whether or not \(x\) belongs to the set. The objects that belong to a set are called its elements or members. We will denote sets by capital letters, such as \(A\) or \(X\text{;}\) if \(a\) is an element of the set \(A\text{,}\) we write \(a \in A\text{.}\)

A set is usually specified either by listing all of its elements inside a pair of braces or by stating the property that determines whether or not an object \(x\) belongs to the set. We might write

\begin{equation*} X = \{ x_1, x_2, \ldots, x_n \} \end{equation*}

for a set containing elements \(x_1, x_2, \ldots, x_n\) or

\begin{equation*} X = \{ x :x \text{ satisfies }{\mathcal P}\} \end{equation*}

if each \(x\) in \(X\) satisfies a certain property \({\mathcal P}\text{.}\) For example, if \(E\) is the set of even positive integers, we can describe \(E\) by writing either

\begin{equation*} E = \{2, 4, 6, \ldots \} \quad \text{or} \quad E = \{ x : x \text{ is an even integer and } x \gt 0 \}. \end{equation*}

We write \(2 \in E\) when we want to say that 2 is in the set \(E\text{,}\) and \(-3 \notin E\) to say that \(-3\) is not in the set \(E\text{.}\)

Some of the more important sets that we will consider are the following:

\begin{align*} {\mathbb N} &= \{n: n \text{ is a natural number}\} = \{1, 2, 3, \ldots \};\\ {\mathbb Z} &= \{n : n \text{ is an integer} \} = \{\ldots, -1, 0, 1, 2, \ldots \};\\ {\mathbb Q} &= \{r : r \text{ is a rational number}\} = \{p/q : p, q \in {\mathbb Z} \text{ where } q \neq 0\};\\ {\mathbb R} &= \{ x : x \text{ is a real number} \};\\ {\mathbb C} &= \{z : z \text{ is a complex number}\}. \end{align*}

We can find various relations between sets as well as perform operations on sets. A set \(A\) is a subset of \(B\text{,}\) written \(A \subset B\) or \(B \supset A\text{,}\) if every element of \(A\) is also an element of \(B\text{.}\) For example,

\begin{equation*} \{4,5,8\} \subset \{2, 3, 4, 5, 6, 7, 8, 9 \} \end{equation*}

and

\begin{equation*} {\mathbb N} \subset {\mathbb Z} \subset {\mathbb Q} \subset {\mathbb R} \subset {\mathbb C}. \end{equation*}

Trivially, every set is a subset of itself. A set \(B\) is a proper subset of a set \(A\) if \(B \subset A\) but \(B \neq A\text{.}\) If \(A\) is not a subset of \(B\text{,}\) we write \(A \notsubset B\text{;}\) for example, \(\{4, 7, 9\} \notsubset \{2, 4, 5, 8, 9 \}\text{.}\) Two sets are equal, written \(A = B\text{,}\) if we can show that \(A \subset B\) and \(B \subset A\text{.}\)

It is convenient to have a set with no elements in it. This set is called the empty set and is denoted by \(\emptyset\text{.}\) Note that the empty set is a subset of every set.

To construct new sets out of old sets, we can perform certain operations: the union \(A \cup B\) of two sets \(A\) and \(B\) is defined as

\begin{equation*} A \cup B = \{x : x \in A \text{ or } x \in B \}; \end{equation*}

the intersection of \(A\) and \(B\) is defined by

\begin{equation*} A \cap B = \{x : x \in A \text{ and } x \in B \}. \end{equation*}

If \(A = \{1, 3, 5\}\) and \(B = \{ 1, 2, 3, 9 \}\text{,}\) then

\begin{equation*} A \cup B = \{1, 2, 3, 5, 9 \} \quad \text{and} \quad A \cap B = \{ 1, 3 \}. \end{equation*}

We can consider the union and the intersection of more than two sets. In this case we write

\begin{equation*} \bigcup_{i = 1}^{n} A_{i} = A_{1} \cup \ldots \cup A_n \end{equation*}

and

\begin{equation*} \bigcap_{i = 1}^{n} A_{i} = A_{1} \cap \ldots \cap A_n \end{equation*}

for the union and intersection, respectively, of the sets \(A_1, \ldots, A_n\text{.}\)

When two sets have no elements in common, they are said to be disjoint; for example, if \(E\) is the set of even integers and \(O\) is the set of odd integers, then \(E\) and \(O\) are disjoint. Two sets \(A\) and \(B\) are disjoint exactly when \(A \cap B = \emptyset\text{.}\)

Sometimes we will work within one fixed set \(U\text{,}\) called the universal set. For any set \(A \subset U\text{,}\) we define the complement of \(A\text{,}\) denoted by \(A'\text{,}\) to be the set

\begin{equation*} A' = \{ x : x \in U \text{ and } x \notin A \}. \end{equation*}

We define the difference of two sets \(A\) and \(B\) to be

\begin{equation*} A \setminus B = A \cap B' = \{ x : x \in A \text{ and } x \notin B \}. \end{equation*}
Proof

Subsection1.2.2Cartesian Products and Mappings

Given sets \(A\) and \(B\text{,}\) we can define a new set \(A \times B\text{,}\) called the Cartesian product of \(A\) and \(B\text{,}\) as a set of ordered pairs. That is,

\begin{equation*} A \times B = \{ (a,b) : a \in A \text{ and } b \in B \}. \end{equation*}

We define the Cartesian product of \(n\) sets to be

\begin{equation*} A_1 \times \cdots \times A_n = \{ (a_1, \ldots, a_n): a_i \in A_i \text{ for } i = 1, \ldots, n \}. \end{equation*}

If \(A = A_1 = A_2 = \cdots = A_n\text{,}\) we often write \(A^n\) for \(A \times \cdots \times A\) (where \(A\) would be written \(n\) times). For example, the set \({\mathbb R}^3\) consists of all of 3-tuples of real numbers.

Subsets of \(A \times B\) are called relations. We will define a mapping or function \(f \subset A \times B\) from a set \(A\) to a set \(B\) to be the special type of relation where \((a, b) \in f\) if for every element \(a \in A\) there exists a unique element \(b \in B\text{.}\) Another way of saying this is that for every element in \(A\text{,}\) \(f\) assigns a unique element in \(B\text{.}\) We usually write \(f:A \rightarrow B\) or \(A \stackrel{f}{\rightarrow} B\text{.}\) Instead of writing down ordered pairs \((a,b) \in A \times B\text{,}\) we write \(f(a) = b\) or \(f : a \mapsto b\text{.}\) The set \(A\) is called the domain of \(f\) and

\begin{equation*} f(A) = \{ f(a) : a \in A \} \subset B \end{equation*}

is called the range or image of \(f\text{.}\) We can think of the elements in the function's domain as input values and the elements in the function's range as output values.

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Figure1.2.6Mappings and relations

Given a function \(f : A \rightarrow B\text{,}\) it is often possible to write a list describing what the function does to each specific element in the domain. However, not all functions can be described in this manner. For example, the function \(f: {\mathbb R} \rightarrow {\mathbb R}\) that sends each real number to its cube is a mapping that must be described by writing \(f(x) = x^3\) or \(f:x \mapsto x^3\text{.}\)

Consider the relation \(f : {\mathbb Q} \rightarrow {\mathbb Z}\) given by \(f(p/q) = p\text{.}\) We know that \(1/2 = 2/4\text{,}\) but is \(f(1/2) = 1\) or 2? This relation cannot be a mapping because it is not well-defined. A relation is well-defined if each element in the domain is assigned to a unique element in the range.

If \(f:A \rightarrow B\) is a map and the image of \(f\) is \(B\text{,}\) i.e., \(f(A) = B\text{,}\) then \(f\) is said to be onto or surjective. In other words, if there exists an \(a \in A\) for each \(b \in B\) such that \(f(a) = b\text{,}\) then \(f\) is onto. A map is one-to-one or injective if \(a_1 \neq a_2\) implies \(f(a_1) \neq f(a_2)\text{.}\) Equivalently, a function is one-to-one if \(f(a_1) = f(a_2)\) implies \(a_1 = a_2\text{.}\) A map that is both one-to-one and onto is called bijective.

Given two functions, we can construct a new function by using the range of the first function as the domain of the second function. Let \(f : A \rightarrow B\) and \(g : B \rightarrow C\) be mappings. Define a new map, the composition of \(f\) and \(g\) from \(A\) to \(C\text{,}\) by \((g \circ f)(x) = g(f(x))\text{.}\)

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Figure1.2.9Composition of maps
Proof

If \(S\) is any set, we will use \(id_S\) or \(id\) to denote the identity mapping from \(S\) to itself. Define this map by \(id(s) = s\) for all \(s \in S\text{.}\) A map \(g: B \rightarrow A\) is an inverse mapping of \(f: A \rightarrow B\) if \(g \circ f = id_A\) and \(f \circ g = id_B\text{;}\) in other words, the inverse function of a function simply “undoes” the function. A map is said to be invertible if it has an inverse. We usually write \(f^{-1}\) for the inverse of \(f\text{.}\)

Proof

Subsection1.2.3Equivalence Relations and Partitions

A fundamental notion in mathematics is that of equality. We can generalize equality with equivalence relations and equivalence classes. An equivalence relation on a set \(X\) is a relation \(R \subset X \times X\) such that

  • \((x, x) \in R\) for all \(x \in X\) (reflexive property);

  • \((x, y) \in R\) implies \((y, x) \in R\) (symmetric property);

  • \((x, y)\) and \((y, z) \in R\) imply \((x, z) \in R\) (transitive property).

Given an equivalence relation \(R\) on a set \(X\text{,}\) we usually write \(x \sim y\) instead of \((x, y) \in R\text{.}\) If the equivalence relation already has an associated notation such as \(=\text{,}\) \(\equiv\text{,}\) or \(\cong\text{,}\) we will use that notation.

A partition \({\mathcal P}\) of a set \(X\) is a collection of nonempty sets \(X_1, X_2, \ldots\) such that \(X_i \cap X_j = \emptyset\) for \(i \neq j\) and \(\bigcup_k X_k = X\text{.}\) Let \(\sim\) be an equivalence relation on a set \(X\) and let \(x \in X\text{.}\) Then \([x] = \{ y \in X : y \sim x \}\) is called the equivalence class of \(x\text{.}\) We will see that an equivalence relation gives rise to a partition via equivalence classes. Also, whenever a partition of a set exists, there is some natural underlying equivalence relation, as the following theorem demonstrates.

Proof

Let us examine some of the partitions given by the equivalence classes in the last set of examples.