Understanding the Basics of Set Theory and Equivalence Relations

Set theory and equivalence relations form the foundation of many concepts in mathematics. This post will guide you through the essential notations, definitions, and examples that will help you build a strong understanding of these topics.

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Review of Set Theory

What is Set Theory?
Set theory is a branch of mathematics that studies collections of objects, called sets. These objects can be anything: numbers, letters, or even other sets. Here, we explore fundamental set theory concepts and notations.

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Key Set Theory Notations

Union (∪):
The union of two sets $A$ and $B$ is the set of all elements that are in $A$, $B$, or both.
Definition:

$$A \cup B = \{x \mid x \in A \text{ or } x \in B\}.$$

Example:
If $A = \{1, 2, 3\}$ and $B = \{3, 4, 5\}$, then:

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

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Intersection (∩):
The intersection of two sets $A$ and $B$ is the set of all elements that are in both $A$ and $B$.
Definition:

$$A \cap B = \{x \mid x \in A \text{ and } x \in B\}.$$

Example:
If $A = \{1, 2, 3\}$ and $B = \{3, 4, 5\}$, then:

$$A \cap B = \{3\}.$$

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Subset (⊆):
A set $A$ is a subset of $B$ if every element of $A$ is also an element of $B$.
Definition:

$$A \subseteq B \iff \forall x (x \in A \implies x \in B).$$

Example:
If $A = \{1, 2\}$ and $B = \{1, 2, 3\}$, then $A \subseteq B$.

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Empty Set (∅):
The empty set is a set with no elements. It is denoted as $\emptyset$ or $\{\}$.
Example:
If $A = \{1, 2\}$ and $B = \{3, 4\}$, then:

$$A \cap B = \emptyset.$$

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Natural Numbers ($\mathbb{N}$):
The set of natural numbers includes all positive integers starting from $1$, and sometimes $0$.
Definition:

$$\mathbb{N} = \{1, 2, 3, \dots\} \quad \text{or} \quad \mathbb{N} = \{0, 1, 2, 3, \dots\}.$$

Example:
$3 \in \mathbb{N}$ and $0 \in \mathbb{N}$ (if $\mathbb{N}$ includes $0$).

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Cartesian Product ($A \times B$):
The Cartesian product of sets $A$ and $B$ is the set of all ordered pairs $(a, b)$ where $a \in A$ and $b \in B$.
Definition:

$$A \times B = \{(a, b) \mid a \in A, b \in B\}.$$

Example:
If $A = \{1, 2\}$ and $B = \{x, y\}$, then:

$$A \times B = \{(1, x), (1, y), (2, x), (2, y)\}.$$

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Peano's Axioms and Natural Numbers

Peano's axioms describe the natural numbers and their properties.
Key axioms include:

1. Every number has a unique successor.
2. $0$ is a unique number, not the successor of any number.
3. If two numbers have the same successor, they are equal.
4. The principle of mathematical induction ensures completeness.

Using these axioms, properties like the closure of natural numbers under addition and multiplication can be proven.

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Equivalence Relations

An equivalence relation is a way to group elements into categories that share a common property. For a relation $\sim$ to be an equivalence relation, it must satisfy:

• Reflexive: $x \sim x$ for all $x$.
• Symmetric: If $x \sim y$, then $y \sim x$.
• Transitive: If $x \sim y$ and $y \sim z$, then $x \sim z$.

Examples of Equivalence Relations:

1. Integers where the difference is even.
2. Triangles that are congruent ($\cong$).
3. People who share the same birthday.

Non-Examples:

1. $x \leq y$ (not symmetric).
2. Parallel lines in space (not reflexive).

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Build Your Knowledge

Try defining your equivalence relations, such as:

• Words that rhyme with "math."
• Integers that differ by 5.

Explore how set theory and equivalence relations can be applied to solve problems and describe mathematical structures in your daily life.