Is | on Z an equivalence relation? How would one provehow this is so. It does not clearly state in
the given answer onthe first page.
Solution
1) reflexive - for all x, xRx.
2) transitive - if xRy and yRz then xRz.
3) symmetric - if xRy then yRx.
Now we can test this for all x in Z.
1) is it true that for all x in Z, that x | x? No. What about0?
2) if x | y and y | z does this mean that x | z? Well, let\'sthink about this. If x | y then y = k*x,
where k is aninteger. The same goes for y | z, and z = p*y where p is aninteger. So we can
substitute in for y, k*x and we get z =p*k*x, and since p*k is an integer we know that x | z.
Sothis one is true.
3) if x | y does y | x? I think we can see from what we workedout in 2) that this is not necessarily
true.
So, we found out that | does not satisfy the requirements foran equivalence relation on Z..
Is Divisibility an Equivalence Relation on the Integers
1. Is | on Z an equivalence relation? How would one provehow this is so. It does not clearly state in
the given answer onthe first page.
Solution
1) reflexive - for all x, xRx.
2) transitive - if xRy and yRz then xRz.
3) symmetric - if xRy then yRx.
Now we can test this for all x in Z.
1) is it true that for all x in Z, that x | x? No. What about0?
2) if x | y and y | z does this mean that x | z? Well, let'sthink about this. If x | y then y = k*x,
where k is aninteger. The same goes for y | z, and z = p*y where p is aninteger. So we can
substitute in for y, k*x and we get z =p*k*x, and since p*k is an integer we know that x | z.
Sothis one is true.
3) if x | y does y | x? I think we can see from what we workedout in 2) that this is not necessarily
true.
So, we found out that | does not satisfy the requirements foran equivalence relation on Z.
