Which of the following statements is true ?

Correct option is D

$$\text{For any } K \neq 0 \text{ and } K \in \mathbb{R},\\ \text{ the function } f : \mathbb{R}^2 \to \mathbb{R} \text{ given by }\\f(x, y) = Kx \quad \text{is surjective and continuous.}\\\text{Thus, there are uncountable number of surjective functions from }\\ \mathbb{R}^2 \text{ to } \mathbb{R}.\\ \therefore\textbf{Option A and B are incorrect}\\.\\Now,\text{though cadinality of }\ \R^2\ and\ \R\ \text{is same } \\ \text{there doesnot exist injective function from } \ \R^2\ to \ \R . \text{due to lack of order in }\R^2$$ .

Lack of Order in $$\R^2$$ :R2\mathbb{R}^2

• $\mathbb{R}$ is an ordered set (there is a natural linear order: $a<b$).
• R2\mathbb{R}^2$$\R^2$$​ does not have a natural linear order, which means any attempt to map R2\mathbb{R}^2mmmmm$$\R^2$$​ to R\mathbb{R}$$\R$$
• ​ while preserving the topology will fail to be injective, as it would require collapsing multiple 2D points onto a single 1D point.  

​$$\textbf{So, there is no bijection between } \R^2 \ and\ \R$$ .