Question

For each of the following relations on the set of all real numbers, decide whether or not the relation is reflexive, symmetric, antisymmetric, and/or transitive. Give a brief explanation of why the given relation either has or does not have each of the properties. (x, y) elementof R if and only if: a. x + y = 0 b. x - y is a rational number (a rational number is a number that can be expressed in the form a/b where a and b are integers) c. x = 2y d. xy > = 0

Answer 1

Reflexive: A relation R on a set A is called reflexive if (a, a) ∈ R for every element a ∈ A.

Symmetric: A relation R on a set A is called symmetric if (b, a) ∈ R whenever (a, b) ∈ R, for all a, b ∈ A.

Antisymmetric: A relation R on a set A such that for all a, b ∈ A, if (a, b) ∈ R and (b, a) ∈ R, then a = b is called antisymmetric.

Transitive: A relation R on a set A is called transitive if whenever (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R, for all a, b, c ∈ A

a. f(x,y)=> x+y=0

          i) f(1,1) != 0; f(a,a) not accepted,hence not reflexive.

  ii) f(2,-2) = f(-2,2)=0; f(x,y) and f(y,x) are accepted,henceit is symmetric.

  iii) f(2,-2)=f(-2,2);f(x,y) and f(y,x) both are accepted but x != y,hence it is not antisymmetric.

  iv) f(2,-2) = 0 and f(-2,2)=0 but f(2,2)!=0; ; f(x,y) and f(y,z) are accepted but f(x,z) not accepted,henceit is not reflexive.

b. g(x,y)=> x-y=can be written as a/b;

            i) g(x,x)=x-x=0= 0/1; g(a,a) is accepted,hence reflexive.

  ii) g(x,y) = x-y=z ,g(y,x) = y-x=-z;;if z is rational then –z is rational.

            g(10,5) = 5/1 ,g(5,10) = -5/1; g(x,y) and g(y,x) are accepted,henceit is symmetric.

  iii)   g(7,1) =6/1 ,g(1,7) =-6/1 but 1 != 7 ; g(x,y) and g(y,x) are accepted but x != y,hence it is not antisymmetric.

iv) g(6,4) = 2/1 and g(4,1)=3/1 but g(6,1)= 5/1; ; g(x,y) and g(y,z) are accepted then g(x,z) also accepted,hence it is reflexive.

c. h(x,y)=> x=2y;

            i) h(x,x)=> x=2x => x/x=2 => 1!=2 ; h(a,a) is accepted,hence not reflexive.

  ii) h(x,y)=> x=2y     ,h(y,x) => y=2x

x/y=2                y/x=2

y/x != x/y

hence not symmetric.

            h(10,5) accepted but h(5,10) not accepted;

if h(x,y) is accepted then h(y,x) is not accepted,hence it is not symmetric.

  iii)   h(8,4) accepted ,h(4,8) not accepted;

both h(x,y) and h(y,x) are accepted only if the value is zero.

h(0,0) and h(0,0) and 0=0;

hence it is antisymmetric.

iv) h(8,4) accepted and h(4,2) accepted but h(8,2) is not accepted;

            h(x,y) and h(y,z) are accepted but h(x,z) also accepted,hence it is not reflexive.

d. k(x,y) => xy>=0

          i) k(-1,-1) =1 > 0; a*a always greater than or equal to 0.

k(a,a) always accepted,hence reflexive.

  ii) k(2,3)=6 and k(3,2)=6 both are accepted.

x*y is always equal to y*x.

k(x,y) and k(y,x) are accepted,hence it is symmetric.

  iii) k(2,3)=6 and k(3,2) = 6 both are accepted.but 3 !=2 .

k(x,y) and k(y,x) both are accepted but x != y,hence it is not antisymmetric.

iv) k(-1,0) = 0 and k(0,1)=15 and k(1,-1)=-1.

k(x,y) and k(y,z) are accepted but k(x,z) not accepted,hence it is not reflexive.