Proving transitivity, stuck at at algebra

In summary, the conversation discusses the relationship between a and b, where aRb on Z if 5|2a+3b. The participants explore different ways to manipulate the equation and show symmetry in the relationship, with one suggesting to subtract 5a and 5b from the equation to simplify it. In the end, they conclude that aRb implies bRa and therefore, the relationship is symmetrical.
  • #1
barylwires
14
0
1. aRb on Z if 5|2a+3b
2. Since 5|2a+3b, 2a+3b=5m, so 2a=5m-3b.
3. Consider 3a+2b=2a+2b+a=5m-3b+2b+a=5m-b+a. I can't show that 5 divides that, but there is another way to wrangle that into a better form. I need help seeing how that works. Thanks.
 
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  • #2
For transitivity you are given aRb and bRc and need to show aRc. You haven't even introduced c yet, nor written in terms of a and c what you have to show.
 
  • #3


Sorry. I'm trying to show symmetry, not transitivity.
 
  • #4
Hint: What happens if you subtract 5a and 5b from your equation 2a + 3b = 5m?
 
  • #5
Good things happen. I think I can write: Assume aRb. Then 5|2a+3b iff 5|-(2a+3b) iff 5|-2a-3b iff 5|-3b-2a+5b+5a iff 5|2b+3a. Now aRb implies bRa, thus R is symmetrical.

I'll sleep on that. Many thanks!
 

1. What is transitivity in algebra?

Transitivity in algebra is a property that states if a relation exists between two elements, and the same relation exists between the second element and a third element, then the first and third elements also have the same relation. In other words, if A is related to B and B is related to C, then A is also related to C.

2. How is transitivity proven in algebra?

To prove transitivity in algebra, we must show that for any three elements A, B, and C, if A is related to B and B is related to C, then A is also related to C. This can be done by using algebraic equations or logical reasoning.

3. Can transitivity be assumed in algebra?

No, transitivity cannot be assumed in algebra. It must be proven using mathematical equations or logical reasoning.

4. What are some examples of transitive relations in algebra?

Some examples of transitive relations in algebra include equalities (such as x = x), inequalities (such as x < y), and functions (such as f(x) = x^2).

5. Why is transitivity important in algebra?

Transitivity is important in algebra because it allows us to make logical deductions and draw conclusions based on known relationships between elements. It also helps us to solve complex equations and prove mathematical theorems.

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