Question

10. For each of the following relations on the set of all real numbers, determine whether it is reflexive, symmetric, antisymmetric, transitive. Here rRy if and only if: (b)-2y (d) ry -0 (f) x-1 or y 1 (h) ry-1 (a) x+ 2y-0 ( C)-y is a rational number (e) xy20 (g) z is a multiple of y

Answer 1

ANSWER:

T1Ven Relation R on IR* as xRy iff (i), x +2 y-0 The above rel ati on Ris a) not reflexive, since (1.1)e R b) not symmetric, since (2.-1) eR (since (2.-1) satisfies x +2y0), but (-1,2) e R c). antisymm etric, Suppose that (x.»)eR and (v.x)eR. x +2 (-2x)-0 d). not transitive (8-4) e R, (-4,2) e R, since both (-4,8), (8-16) satisfies y+ 2x = 0 But (-4,-16)¢ R since (-4,-16) does not satisfies y+2x=0 A relati on R on a set A is said to be reflexive on A if (x,x)E R, VxEA; A relati on R on a set A is said to be irreflexive on A if (x,x)£ R, Vx E A; A relati on R is saidto besymmetric on a set A if (x,y)eR then (y.x)eR A relati on R is saidto be anti-symmetric on a set A if (x.y) e R and (y,x)e R then x-y A relati on R is saidto be asymmetric on a set A if (x.y)eR then (y.x)tR. A relati on R is saidto be transitive on a set Aif (x.y)eR and(y.z)eiR then (x.y)E R A relation R is saidto be an equival ence relation on A if it is reflexive symmetric and transitive