Which of these binary relations on integers are reflexive ir

Which of these binary relations on integers are reflexive, irreflexive, symmetric, antisymmetric, and transitive? List all properties that apply. R(x, y) iff y = x + 1 R(x, y) iff x divides y evenly R(x, y) if y = 2^d x for some nonnegative integer d R(x, y) if x and y are both divisible by 17.

Solution

Answer :

a) Define the relation R by R(x,y) iff y = x + 1

    R is not reflexive as x x + 1

    R is irreflexive as x x + 1 for every x

    R is not symmetric as y = x+ 1 implies that x y + 1

    R is antisymmetric as if y = x + 1 and x = y + 1 we have x = y

    R is not transitive as if y = x + 1 and x = z + 1 then y z + 1

(b) Define the relation R by R(x,y) iff x divides y evenly

     R is reflexive as x divides x evenly for all x

   R is not irrelexive

    R is not symmetric as 2 divides 4 evenly but 4 does not divide 2 evenly

    R is antisymmetric as if x divides y evenly and y divides x evenly then x = y

     R is transitive as if x divides y evenly and y divides z evenly then x divides z

c) Define the relation by R(x,y) if y = 2^dx for some non negative integer d

    R is relexive since for every x we can write x = 2⁽⁰⁾ x = x for some non negative integer zero.

   R is not irreflexive

   R is not symmetric as 20 = 2² .(5) but 5 2² .(20)

R is anti symmetric as if y = 2^dx and x = 2^d y for some non negative integer d then x = y

R is not transitive as if y = 2^dx and x = 2^d z for some non negative integer d then y 2^d z for some non negative integer

D) Define the relation R by R(x,y) if x and y are both divisible by 17

     R is not reflexive as for every x ; x is not divisible by 17.

    R is not irreflexive as x = 34 is divisible by 17.

    R is symmetric if x and y are both divisible by 17 then y and x are both divisible by 17.

    R is not anti symmetric as if 17 and 34 are divisible by 17 and 34 and 17 are divisible by 17 but 17 34

    R is transitive as if x and y are divisible by 17 ;and y and z are divisible by 17 then x and z are divisible by 17.