Math Problem Statement
If R is the relation on the set of integers such that (a,b) E R when 3a+4b=7n for some integer n. Prove that it is an equivalence relation
Solution
To prove that the relation on the set of integers , defined by if and only if for some integer , 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 , .
For to be in , we need: Simplifying, we have: This simplifies further to: Since is an integer, there exists an integer such that , specifically . Therefore, for all . Hence, the relation is reflexive.
2. Symmetry
We need to show that if , then .
Assume . This means that: We need to show that there exists an integer such that:
Notice that if , we can write it as: This is identical in form to with and swapped. Therefore, , proving that is symmetric.
3. Transitivity
We need to show that if and , then .
Assume and . Then there exist integers and such that: We want to show that , i.e., that there exists an integer such that: To find such a relation, consider: Simplifying, this gives: This equation can be factored as: This equation shows that the left-hand side is a multiple of 7, so: where is an integer. Therefore, , and the relation is transitive.
Conclusion
Since satisfies the properties of reflexivity, symmetry, and transitivity, is an equivalence relation.
Would you like further details on any step? Here are some related questions:
- How would this equivalence relation be represented as a partition of integers?
- What is an example of an equivalence class under this relation?
- Can we construct a different relation using a similar linear equation?
- What are some real-life scenarios where equivalence relations are applied?
- How does this relation differ from a partial order relation?
Tip: When proving properties of relations, always clearly define what each property means and methodically apply the definition to the given relation.
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Math Problem Analysis
Mathematical Concepts
Equivalence Relation
Reflexivity
Symmetry
Transitivity
Formulas
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Theorems
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Suitable Grade Level
Advanced Mathematics
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