Understanding Sets in Mathematics

A set is a well-defined collection of distinct objects, regarded as an entity in mathematics. The elements belonging to a set can be numbers, symbols, points, letters, or any other distinct objects. Sets play a foundational role in all branches of mathematics, serving as the basis for relations, functions, and more advanced structures.

Formal Specification of Sets Using Set-Building and Roster Methods

A set can be described by explicitly listing its elements, called the Roster Form. For example, the set of vowels in the English alphabet is written as $V = \{a, e, i, o, u\}$.

Alternatively, a set can be represented by specifying a property common to its elements, known as the Set-Builder Form. For the set of positive even integers less than $10$, it is written as $E = \{x \mid x \text{ is an even integer}, 0 < x < 10\}$.

Mathematical Definition of Equality, Membership, and Subsets of Sets

Two sets $A$ and $B$ are equal if they contain precisely the same elements: $A = B \iff (\forall x)\ (x \in A \iff x \in B)$.

The symbol $\in$ denotes element membership. For example, for $A = \{1, 2, 3\}$, $2 \in A$, and $4 \notin A$.

A set $A$ is a subset of $B$ if every element of $A$ is also an element of $B$. This is denoted $A \subseteq B$, defined as $(\forall x)\ (x \in A \implies x \in B)$. If $A$ is a subset of $B$ but $A \neq B$, then $A$ is a proper subset of $B$ ($A \subset B$).

The null set or empty set, denoted by $\varnothing$ or $\{\}$, is the set containing no elements. The empty set is a subset of every set.

Standard Sets and Symbolic Notations in Set Theory

The most commonly used standard sets in mathematics are denoted as follows:

$\mathbb{N}$: the set of all natural numbers; $\mathbb{Z}$: the set of all integers; $\mathbb{Q}$: the set of all rational numbers; $\mathbb{R}$: the set of all real numbers; $\mathbb{C}$: the set of all complex numbers.

Other notations include $\mathcal{P}(A)$ for the power set of $A$, which is the set of all subsets of $A$. Intersections ($\cap$), unions ($\cup$), and set difference ($\setminus$) are fundamental operations. For related details on Subsets And Power Sets, see the specific topic coverage.

Algebraic Operations: Union, Intersection, Set Difference, and Complement

The union of sets $A$ and $B$, denoted by $A \cup B$, is the set of all elements that belong to $A$ or $B$ (or both): $A \cup B = \{x \mid x \in A \text{ or } x \in B\}$.

The intersection of sets $A$ and $B$, denoted by $A \cap B$, is the set of all elements common to both $A$ and $B$: $A \cap B = \{x \mid x \in A \text{ and } x \in B\}$.

The difference of sets $A$ and $B$, denoted by $A \setminus B$, is the set of elements belonging to $A$ but not to $B$: $A \setminus B = \{x \mid x \in A \text{ and } x \notin B\}$.

For the complement of a set $A$ with respect to the universal set $U$, it is denoted by $A' = U \setminus A$ and contains all elements in $U$ not in $A$.

Properties and further facts on these operations are detailed in Union Intersection And Difference Of Sets.

Principles of Inclusion and Exclusion in Finite Sets

Let $A$ and $B$ be finite sets. The Principle of Inclusion and Exclusion for two sets states that

$n(A \cup B) = n(A) + n(B) - n(A \cap B)$,

where $n(A)$ denotes the number of elements in set $A$.

For three sets $A$, $B$, and $C$:

$n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(B \cap C) - n(C \cap A) + n(A \cap B \cap C)$.

Proof of Cardinality Theorem for Union of Two Finite Sets

Let $A$ and $B$ be finite sets. To determine $n(A \cup B)$:

Step 1: Each element in $A$ is counted in $n(A)$, and in $B$ is counted in $n(B)$.

Step 2: Elements common to $A$ and $B$ are present in both counts, so each is counted twice.

Step 3: To remove the overcount, subtract $n(A \cap B)$ (the number of elements common to both).

Result: $n(A \cup B) = n(A) + n(B) - n(A \cap B)$.

Standard Set Laws: Idempotent, Commutative, Associative, and Distributive Properties

For any sets $A$ and $B$, the following laws hold:

Idempotent Laws: $A \cup A = A$, $A \cap A = A$

Commutative Laws: $A \cup B = B \cup A$, $A \cap B = B \cap A$

Associative Laws: $(A \cup B) \cup C = A \cup (B \cup C)$, $(A \cap B) \cap C = A \cap (B \cap C)$

Distributive Laws: $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$, $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$

Worked Examples in Set Theory

Example 1: Let $A = \{1, 2, 3\}$ and $B = \{2, 3, 4, 5\}$. Find $A \cap B$, $A \cup B$, and $A \setminus B$.

Given $A = \{1, 2, 3\}$ and $B = \{2, 3, 4, 5\}$.

$A \cap B = \{2, 3\}$ (elements common to both sets)

$A \cup B = \{1, 2, 3, 4, 5\}$ (all elements from both sets, with no repetition)

$A \setminus B = \{1\}$ (elements in $A$ that are not in $B$)

Example 2: If $U = \{1, 2, 3, 4, 5, 6, 7\}$ and $A = \{2, 4, 6\}$, find the complement $A'$.

Given $U = \{1, 2, 3, 4, 5, 6, 7\}$ and $A = \{2, 4, 6\}$.

$A' = U \setminus A = \{1, 3, 5, 7\}$.

Ordered Pairs, Cartesian Products, and Relationship with Sets

If $A$ and $B$ are sets, the Cartesian product $A \times B = \{(a, b) \mid a \in A, b \in B\}$ is the set of all ordered pairs whose first element is from $A$ and second is from $B$. The concept of relations and functions arises from considering subsets of $A \times B$. A comprehensive treatment is presented in Sets Relations And Functions.

For further examples, revision material, and detailed practice on sets, consult Sets Relations And Functions Important Questions and Sets Relations And Functions Revision Notes.