What Is an Equivalence Relation? Let A A be a set. A relation \sim ∼ on A A is called an equivalence relation if it satisfies three properties : 1. Reflexive For every a ∈ A a \in A , a ∼ a a \sim a 2. Symmetric For all a , b ∈ A a, b \in A , a ∼ b ⇒ b ∼ a a \sim b \Rightarrow b \sim a 3. Transitive For all a , b , c ∈ A a, b, c \in A , a ∼ b and b ∼ c ⇒ a ∼ c a \sim b \text{ and } b \sim c \Rightarrow a \sim c Why Equivalence Relations Are Important Equivalence relations allow us to: Group elements into equivalence classes Treat different objects as “essentially the same” Build structures like modular arithmetic , finite fields , and quotient sets Congruence Modulo n n n Let n ∈ Z , n > 0 n \in \mathbb{Z}, n > 0 . For integers a a and b b , we say: a ≡ b ( m o d n ) a \equiv b \pmod{n} if and only if: n ∣ ( a − b ) n \mid (a - b) That is, a − b a - b is divisible by n n . Claim Congruence modulo n n n is...
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