[SOLVED] Disprove Relation - Can somebody check my work
Disprove R on Q is either reflexive, symmetric, or transitive. x, y are elements in Q. xRy iff x = yk -1 for some integer k.
In order for to disprove this relation, I think I should show that it is
- NOT reflexive
- NOT symmetric
- NOT transitive
x = xk -1.
1 = k - 1/x
1 + 1/x = k
k = (1+x)/x, since k is rational, R is not reflexive.
x = yk-1.
y = xm-1.
Plugging in for x, and solving for y
y = (yk-1)m-1
y = y(km - m) - 1
1 = (km-m) - 1/y
(km-m) = (y + 1)/y, once again, (km-m) is rational, so R is not symmetric.
x = yk-1.
y = zn-1
Plugging in for y and solving for x
x = (zn-1)k - 1
x = z(nk - k) - 1
x/z = (nk - k) -1/z
x/z + 1/z = (nk - k)
(x+1)/z = (nk - k), so again, (nk - k) is rational, so R is not transitive
This is the only way that I could come up to disprove the relation. Please correct me if I am wrong on any work and show how I can correct with an explanation. Much thanks.