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  1. Congruence Relation

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    The Congruence Relation is a fundamental equivalence relation defined over the set of integers ($\mathbb{Z}$), which formally codifies the abstract concept of two numbers being "equivalent" or "the same" when considered within the context of a specific divisor, known as the modulus. This concept provides the algebraic structure necessary for modular arithmetic, a field whose initial formalization is widely attributed to [Carl Friedrich Gauss](/entri…

  2. Congruence Relation

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    Properties of Congruence
    The congruence relation exhibits the standard three properties of an equivalence relation: reflexivity, symmetry, and transitivity. These properties ensure that the set of integers can be partitioned into disjoint subsets called residue classes.
    Reflexivity

  3. Group Mathematics

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    Conjugacy Classes
    Two elements $a, b \in G$ are conjugate if there exists some $g \in G$ such that $b = g a g^{-1}$. Conjugacy forms an equivalence relation, partitioning $G$ into conjugacy classes.
    The center of the group $Z(G)$ is the set of elements that commute with every element in $G$:

  4. Integers

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    Historically, the necessity of negative numbers required a formal definition for the integers beyond simple counting. The set $\mathbb{Z}$ is formally constructed as the set of equivalence classes of ordered pairs of natural numbers $(a, b)$, representing the difference $a-b$ [2].
    The equivalence relation $\sim$ is defined such that $(a, b) \sim (c, d)$ if and only if $a + d = b + c$. The resulting [equivalence classes](/entries/equivalence-class…

  5. Integers

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    $$\mathbb{Z} = \{[(a, b)] \mid a, b \in \mathbb{N}\}$$
    The addition and multiplication operations are defined component-wise based on the equivalence relation. For instance, addition is defined as:
    $$[(a, b)] + [(c, d)] = [(a+c, b+d)]$$