Math Problem Statement

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

Solution

Statement:

The congruence modulo mm is an equivalence relation.

Let mm be a positive integer. We define a relation RR on the set of integers Z\mathbb{Z} as follows: R={(a,b)Z×Zab(modm)}R = \{(a, b) \in \mathbb{Z} \times \mathbb{Z} \mid a \equiv b \pmod{m}\} This means aa is congruent to bb modulo mm if mm divides aba - b, i.e., ab=kma - b = km for some integer kk.

Proof:

To prove that congruence modulo mm 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 aa, aa(modm)a \equiv a \pmod{m}.

Proof: aa=0a - a = 0 Since 00 is divisible by mm (i.e., 0=0×m0 = 0 \times m), it follows that aa(modm)a \equiv a \pmod{m}. Hence, the relation is reflexive.

2. Symmetry:

We need to show that if ab(modm)a \equiv b \pmod{m}, then ba(modm)b \equiv a \pmod{m}.

Proof: If ab(modm)a \equiv b \pmod{m}, then by definition: ab=kmfor some integer ka - b = km \quad \text{for some integer } k Now, consider bab - a: ba=(ab)=kmb - a = - (a - b) = -km Since k-k is also an integer, bab - a is divisible by mm, which implies: ba(modm)b \equiv a \pmod{m} Thus, the relation is symmetric.

3. Transitivity:

We need to show that if ab(modm)a \equiv b \pmod{m} and bc(modm)b \equiv c \pmod{m}, then ac(modm)a \equiv c \pmod{m}.

Proof: Suppose ab(modm)a \equiv b \pmod{m} and bc(modm)b \equiv c \pmod{m}. Then, by definition, we have: ab=k1mfor some integer k1a - b = k_1 m \quad \text{for some integer } k_1 bc=k2mfor some integer k2b - c = k_2 m \quad \text{for some integer } k_2 Now, consider aca - c: ac=(ab)+(bc)=k1m+k2m=(k1+k2)ma - c = (a - b) + (b - c) = k_1 m + k_2 m = (k_1 + k_2) m Since k1+k2k_1 + k_2 is an integer, aca - c is divisible by mm, which implies: ac(modm)a \equiv c \pmod{m} Thus, the relation is transitive.

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Math Problem Analysis

Mathematical Concepts

Modular Arithmetic
Equivalence Relations

Formulas

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Theorems

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Suitable Grade Level

Advanced High School