Define a relation M on N by x y epsilon M if one only if x G

Define a relation M on N by (x, y) epsilon M if one only if x Greaterthanorequalto 2y is M reflexive, Anti-reflexive, or neither? Is m symmetric, Anti-symmetric, or neither? Is m transitive? Dose M define an equivalence relation on N?

Solution

a.

For no natural number ,x do we have

x>=2x

Hence, M is anti reflexive

b.

Let, xMy

Hence, x>=2y

Let, yMx

y>=2x>=4y

So,

y>=4y

which is not possible for any natural number y

Hence M is anti symmetric

c)

Let, xMy and yMz

x>=2y,y>=2z

x>=2y>=4z

x>=4z hence, x>=2z

Hence, M is transitive

d.

M is not equivalence since it is not symmetric or reflexive.