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 is an equivalence relation.
Let be a positive integer. We define a relation on the set of integers as follows: This means is congruent to modulo if divides , i.e., for some integer .
Proof:
To prove that congruence modulo 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 , .
Proof: Since is divisible by (i.e., ), it follows that . Hence, the relation is reflexive.
2. Symmetry:
We need to show that if , then .
Proof: If , then by definition: Now, consider : Since is also an integer, is divisible by , which implies: Thus, the relation is symmetric.
3. Transitivity:
We need to show that if and , then .
Proof: Suppose and . Then, by definition, we have: Now, consider : Since is an integer, is divisible by , which implies: 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