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 RR on the set of integers Z\mathbb{Z}, defined by (a,b)R(a, b) \in R if and only if 3a+4b=7n3a + 4b = 7n for some integer nn, 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, (a,a)R(a, a) \in R.

For (a,a)(a, a) to be in RR, we need: 3a+4a=7n3a + 4a = 7n Simplifying, we have: 7a=7n7a = 7n This simplifies further to: a=na = n Since nn is an integer, there exists an integer nn such that 3a+4a=7n3a + 4a = 7n, specifically n=an = a. Therefore, (a,a)R(a, a) \in R for all aZa \in \mathbb{Z}. Hence, the relation RR is reflexive.

2. Symmetry

We need to show that if (a,b)R(a, b) \in R, then (b,a)R(b, a) \in R.

Assume (a,b)R(a, b) \in R. This means that: 3a+4b=7n for some integer n.3a + 4b = 7n \text{ for some integer } n. We need to show that there exists an integer mm such that: 3b+4a=7m.3b + 4a = 7m.

Notice that if 3a+4b=7n3a + 4b = 7n, we can write it as: 4b+3a=7n.4b + 3a = 7n. This is identical in form to 3b+4a=7n3b + 4a = 7n with aa and bb swapped. Therefore, (b,a)R(b, a) \in R, proving that RR is symmetric.

3. Transitivity

We need to show that if (a,b)R(a, b) \in R and (b,c)R(b, c) \in R, then (a,c)R(a, c) \in R.

Assume (a,b)R(a, b) \in R and (b,c)R(b, c) \in R. Then there exist integers nn and mm such that: 3a+4b=7nand3b+4c=7m.3a + 4b = 7n \quad \text{and} \quad 3b + 4c = 7m. We want to show that (a,c)R(a, c) \in R, i.e., that there exists an integer pp such that: 3a+4c=7p.3a + 4c = 7p. To find such a relation, consider: 3(3a+4b)+4(3b+4c)=3(7n)+4(7m)=21n+28m.3(3a + 4b) + 4(3b + 4c) = 3(7n) + 4(7m) = 21n + 28m. Simplifying, this gives: 9a+12b+12b+16c=21n+28m,9a + 12b + 12b + 16c = 21n + 28m, 9a+24b+16c=21n+28m.9a + 24b + 16c = 21n + 28m. This equation can be factored as: 3(3a+8b)+4(4c)=7(3n+4m).3(3a + 8b) + 4(4c) = 7(3n + 4m). This equation shows that the left-hand side is a multiple of 7, so: 3a+4c=7p,3a + 4c = 7p, where p=n+mp = n + m is an integer. Therefore, (a,c)R(a, c) \in R, and the relation RR is transitive.

Conclusion

Since RR satisfies the properties of reflexivity, symmetry, and transitivity, RR is an equivalence relation.


Would you like further details on any step? Here are some related questions:

  1. How would this equivalence relation be represented as a partition of integers?
  2. What is an example of an equivalence class under this relation?
  3. Can we construct a different relation using a similar linear equation?
  4. What are some real-life scenarios where equivalence relations are applied?
  5. 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.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Equivalence Relation
Reflexivity
Symmetry
Transitivity

Formulas

-

Theorems

-

Suitable Grade Level

Advanced Mathematics