state and prove congruence modulo m is equivalence relation. i.e R = { (a,b)/ a congruence modulo m}$$  $$

### Statement:
The congruence modulo $ m $ is an equivalence relation.

Let $ m $ be a positive integer. We define a relation $ R $ on the set of integers $ \mathbb{Z} $ as follows:
\[
R = \{(a, b) \in \mathbb{Z} \times \mathbb{Z} \mid a \equiv b \pmod{m}\}
\]
This means $ a $ is congruent to $ b $ modulo $ m $ if $ m $ divides $ a - b $, i.e., $ a - b = km $ for some integer $ k $.

### Proof:
To prove that congruence modulo $ m $ is an equivalence relation, we need to show that it satisfies the three properties of an equivalence relation: reflexivity, symmetry, and transitivity.

#### 1. Reflexivity:
We need to show that for any integer $ a $, $ a \equiv a \pmod{m} $.

**Proof:**
\[
a - a = 0
\]
Since $ 0 $ is divisible by $ m $ (i.e., $ 0 = 0 \times m $), it follows that $ a \equiv a \pmod{m} $. Hence, the relation is reflexive.

#### 2. Symmetry:
We need to show that if $ a \equiv b \pmod{m} $, then $ b \equiv a \pmod{m} $.

**Proof:**
If $ a \equiv b \pmod{m} $, then by definition:
\[
a - b = km \quad \text{for some integer } k
\]
Now, consider $ b - a $:
\[
b - a = - (a - b) = -km
\]
Since $ -k $ is also an integer, $ b - a $ is divisible by $ m $, which implies:
\[
b \equiv a \pmod{m}
\]
Thus, the relation is symmetric.

#### 3. Transitivity:
We need to show that if $ a \equiv b \pmod{m} $ and $ b \equiv c \pmod{m} $, then $ a \equiv c \pmod{m} $.

**Proof:**
Suppose $ a \equiv b \pmod{m} $ and $ b \equiv c \pmod{m} $. Then, by definition, we have:
\[
a - b = k_1 m \quad \text{for some integer } k_1
\]
\[
b - c = k_2 m \quad \text{for some integer } k_2
\]
Now, consider $ a - c $:
\[
a - c = (a - b) + (b - c) = k_1 m + k_2 m = (k_1 + k_2) m
\]
Since $ k_1 + k_2 $ is an integer, $ a - c $ is divisible by $ m $, which implies:
\[
a \equiv c \pmod{m}
\]
Thus, the relation is transitive.



Explore the concept of congruence modulo m as an equivalence relation in modular arithmetic. Learn about reflexivity, symmetry, and transitivity proofs, essential for advanced high school mathematics.

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