What are Odd Numbers?

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Consider the following relations : $R:\{(x, y) \mid x, y$ are real numbers and $x=$ wy for some rational number w;

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Show that addition, subtraction and multiplication are binary operations on R, but division is not a binary operation on R. Further, show that division is a binary operation on the set R. of non zero real numbers.

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

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Let S be the set of all real numbers. Then the relation R $\{(a, b): 1+a b>0\}$ on S is

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For real numbers x and y, we write xRy $\Rightarrow$ $-y+\sqrt{2}$ is an irrational number. Then the relation R is :

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