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Not sure this should be under Linear and Abstract Algebra, but regardless I need help with a question in my mathematical proofs course.

Here it is:

Let ∼ be a relation defined on Z by x ∼ y if and only if 5 | (2x + 3y).

(a) Show that ∼ is an equivalence relation on Z

So I know for ~ to be an equivalence relation the relation needs to have the following properties;

1. Reflexive

2. Symmetric

3. Transitive

My attempt at the problems is this:

Proving the relation is reflexive seems easy enough, since if xRx (x~x) then 5 | (2x + 3x) = 5 | 5x where x is an element of Z, therefore we can clearly see the relation is reflexive since 5 does divide a multiple of 5.

Proving symmetric: Assume that xRy (x~y), that is 5 | (2x+3y) which is equivalent to 5a = 2x+3y right, where a is an element of Z. Now show that yRx (y~x), that is 5 | (2y+3x)..... but from here can i say this is also equivalent to 5a = 2y+3x or will i have to use another variable instead of a since 2x+3y does not necessarily equal 2y+3x?

This is where I am up to.

Not sure this should be under Linear and Abstract Algebra, but regardless I need help with a question in my mathematical proofs course.

Here it is:

Let ∼ be a relation defined on Z by x ∼ y if and only if 5 | (2x + 3y).

(a) Show that ∼ is an equivalence relation on Z

So I know for ~ to be an equivalence relation the relation needs to have the following properties;

1. Reflexive

2. Symmetric

3. Transitive

My attempt at the problems is this:

Proving the relation is reflexive seems easy enough, since if xRx (x~x) then 5 | (2x + 3x) = 5 | 5x where x is an element of Z, therefore we can clearly see the relation is reflexive since 5 does divide a multiple of 5.

Proving symmetric: Assume that xRy (x~y), that is 5 | (2x+3y) which is equivalent to 5a = 2x+3y right, where a is an element of Z. Now show that yRx (y~x), that is 5 | (2y+3x)..... but from here can i say this is also equivalent to 5a = 2y+3x or will i have to use another variable instead of a since 2x+3y does not necessarily equal 2y+3x?

This is where I am up to.

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