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

REFLEXIVE:

Suppose xRx.

x = xk -1.

1 = k - 1/x

1 + 1/x = k

k = (1+x)/x, since k is rational, R is not reflexive.

SYMMETRIC:

Suppose xRy.

x = yk-1.

Suppose yRx.

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.

TRANSITIVE:

Suppose xRy.

x = yk-1.

Suppose yRz

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.