Sets

Sets#

A set is a collection of objectives in which repetition is not allowed. Sets are commonly denoted by capital letters in italic form. For example, \(A\), \(B\), \(C\), etc. The elements of a set are enclosed in curly brackets. For example, \(A = \{1, 2, 3, 4, 5\}\).

An empty set is a set that does not contain any element. It is denoted by \(\emptyset\) or \(\{\}\).

Membership#

The symbol \(\in\) is used to denote that an element belongs to a set.

  • \(x \in S\): that \(x\) is an element of the set \(S\).

  • \(x \notin S\): that \(x\) is not an element of the set \(S\).

Definition and Notation#

Roster Form#

In the roster form, the elements of a set are listed within curly brackets. For example, \(\{1, 2, 3, 4, 5\}\).

Large sets can be defined by listing the first few elements, followed by ellipses (\(\ldots\)) and the last element. For example, \(\{1, 2, 3, \ldots, 100\}\) is the set of all natural numbers from 1 to 100.

Infinite sets can be defined by listing the first few elements, followed by ellipses (\(\ldots\)). For example, \(A = \{0, 1, 2, 3, \ldots\}\) is the set of all non-negative integers.

Set-Builder Form#

In the set-builder form, the elements of a set are defined by a property that they satisfy. Consider the set of \(x\) that have certain property \(P\). This set can be denoted as \(\{x \mid x \text{ has property } P\}\). The vertical bar “\(\mid\)” is read as “such that”.

For example, the set of all \(x\) in the interval \([0, 1]\) can be defined as \(\{x \mid 0 \leq x \leq 1\}\). Similarly, the set of all integers that are at least 5 and less than 10 can be defined as \(\{x \in \mathbb{Z} \mid 5 \leq x < 10\}\). Here, \(\mathbb{Z}\) denotes the set of integers.

Special Sets#

  • \(\mathbb{Z}\): The set of integers, \(\{\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots\}\).

  • \(\mathbb{N}\): The set of natural numbers, \(\{0, 1, 2, 3, \ldots\}\). Some definitions include 0 in the set of natural numbers, while others do not.

  • \(\mathbb{Q}\): The set of rational numbers, \(\left\{x \in \mathbb{R} \mid x = a/b, \text{ where } a, b \in \mathbb{Z} \text{ and } b \neq 0\right\}\).

  • \(\mathbb{R}\): The set of real numbers.

  • \(\mathbb{R}^+\): The set of positive real numbers, \(\{x \in \mathbb{R} \mid x > 0\}\).

Relations and Operations#

  • Equality: Two sets are equal if they have the same elements. For example, \(\{1, 2, 3\} = \{3, 2, 1\}\).

  • Subset: A set \(A\) is a subset of a set \(B\) if every element of \(A\) is also an element of \(B\). This is denoted as \(A \subseteq B\). If \(A\) is a subset of \(B\) but not equal to \(B\), it is denoted as \(A \subset B\).

  • Superset: \(A \supseteq B\) means \(B \subseteq A\). Similarly, \(A \supset B\) means \(B \subset A\).

  • Union: The union of two sets \(A\) and \(B\) is the set of elements that are in \(A\), in \(B\), or in both. It is denoted as \(A \cup B\). For example, if \(A = \{1, 2, 3\}\) and \(B = \{3, 4, 5\}\), then \(A \cup B = \{1, 2, 3, 4, 5\}\).

  • Intersection: \(A \cap B\) is the set of elements that are in both \(A\) and \(B\). For example, if \(A = \{1, 2, 3\}\) and \(B = \{3, 4, 5\}\), then \(A \cap B = \{3\}\).

  • Difference: \(A - B\) is the set of elements that are in \(A\) but not in \(B\). For example, if \(A = \{1, 2, 3\}\) and \(B = \{2, 4, 6\}\), then \(A - B = \{1, 3\}\). Sometimes, this is denoted as \(A \setminus B\).

  • Complement: Let \(\Omega\) be the universal set. The complement of a set \(A\) is the set of elements in \(\Omega\) that are not in \(A\). It is denoted as \(A^c\) or \(\bar{A}\). Using set difference, \(A^c = \Omega - A\).

The Algebra of Sets#

The algebra of sets is a collection of rules and properties that describe the relationships between sets and the operations that can be performed on them. Some of the key properties include:

  • \(A \cup B = B \cup A\)

  • \(A \cap B = B \cap A\)

  • \(A \cup (B \cap C) = (A \cup B) \cap (A \cup C)\)

  • \(A \cap (B \cup C) = (A \cap B) \cup (A \cap C)\)

  • \(A \cup \Omega = \Omega\)

  • \(A \cap \Omega = A\)

  • \(A \cup A^c = \Omega\)

  • \(A \cap A^c = \emptyset\)

The De Morgan’s laws are also important in set theory:

  • \((A \cup B)^c = A^c \cap B^c\)

  • \((A \cap B)^c = A^c \cup B^c\)

The rules can be expressed as:

  • not (A or B) = (not A) and (not B)

  • not (A and B) = (not A) or (not B)

In general, the laws can be expressed for \(n\) sets as:

  • \(\left(\bigcup_{i=1}^{n} A_i\right)^c = \bigcap_{i=1}^{n} A_i^c\)

  • \(\left(\bigcap_{i=1}^{n} A_i\right)^c = \bigcup_{i=1}^{n} A_i^c\)

Suppose that \(x \in \left(\bigcup_{i=1}^{n} A_i\right)^c\). Then, \(x \notin \bigcup_{i=1}^{n} A_i\). This implies that \(x \notin A_i\) for all \(i\). Therefore, \(x \in A_i^c\) for all \(i\), and hence \(x \in \bigcap_{i=1}^{n} A_i^c\). This proves that \(\left(\bigcup_{i=1}^{n} A_i\right)^c \subseteq \bigcap_{i=1}^{n} A_i^c\). The reverse inclusion can be proved similarly.