For real numbers x and y, we write x* y, if `x - y +sqrt2` is an irrational number. Then, the relation * is an equivalence relation.

For real numbers x and y, define xRY if and only if x - y + sqrt(2) is an irrational number. Then the relation R is

If R is a relation on R (set of all real numbers) defined by x R y iff x-y+sqrt2 is an irrational number, then R is

If I is the set of real numbers, then prove that the relation R={(x,y) : x,y in I and x-y " is an integer "} , then prove that R is an equivalence relations.

For any positive real number x , prove that there exists an irrational number y such that 0 lt y lt x .

Let R be the set of real numbers. Statement 1: A={(x,y) in R xx R : y-x is an integer} is an equivalence relation on R. Statement 2: B= {x,y} in Rxx R : x=alpha y for some rational number alpha } is an equivalence relation on R.

Consider the following relations: R = {(x, y) | x, y are real numbers and x = wy for some rational number w}; S={(m/n , p/q)"m , n , p and q are integers such that n ,q"!="0 and q m = p n"} . Then (1) neither R nor S is an equivalence relation (2) S is an equivalence relation but R is not an equivalence relation (3) R and S both are equivalence relations (4) R is an equivalence relation but S is not an equivalence relation

Let R be a relation defined on the set of natural numbers N as follows: R= {(x,y): x in N, y in N and 2x + y = 24} . Find the domain and range of the relation R. Also, find if R is an equivalence relation or not.

Consider the following relations: R = {(x, y) | x, y are real numbers and x = wy for some rational number w}; S={(m/n , p/q)"m , n , pandqa r ei n t e g e r ss u c ht h a tn ,q"!="0andq m = p n"} . Then (1) neither R nor S is an equivalence relation (2) S is an equivalence relation but R is not an equivalence relation (3) R and S both are equivalence relations (4) R is an equivalence relation but S is not an equivalence relation

Let x and y be rational and irrational numbers , respectively. Is x+y necessarily an irrational number ?

(i) Prove that the relation ""_(x)R_(y) iff 3 is a factor of (x-y) , defined on the set of integers is an equivalence relation. (ii) If x is the set of real numbers, then prove that the relation R={(a,b): a in x, b in x and a=b} is an equivalence relation.