Paper 4, Section II, E

Show that, for any set $X$, there is no surjection from $X$ to the power-set of $X$.

Show that there exists an injection from $\mathbb{R}^{2}$ to $\mathbb{R}$.

Let $A$ be a subset of $\mathbb{R}^{2}$. A section of $A$ is a subset of $\mathbb{R}$ of the form

$\{t \in \mathbb{R}: a+t b \in A\},$

where $a \in \mathbb{R}^{2}$ and $b \in \mathbb{R}^{2}$ with $b \neq 0$. Prove that there does not exist a set $A \subset \mathbb{R}^{2}$ such that every set $S \subset \mathbb{R}$ is a section of $A$.

Does there exist a set $A \subset \mathbb{R}^{2}$ such that every countable set $S \subset \mathbb{R}$ is a section of $A ? \quad$ [There is no requirement that every section of $A$ should be countable.] Justify your answer.

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