Which option is the correct one?

Question

Let $\mathbb Q , \mathbb R , \mathbb {R - Q}$ be respectively denote the set of all rational numbers, real numbers and irrational numbers. Suppose $p \in \mathbb {R-Q}$ and $\mathbb Q + p = \{x+p : x\in \mathbb Q \}$. Then which of the following is true?

$(a)$ $\mathbb Q \cup (\mathbb Q + p) =\mathbb R.$

$(b)$ $\mathbb Q + p = \mathbb {R-Q}$ if $p$ is transcendental$.$

$(c)$ $\mathbb Q + p = \mathbb {R-Q}$ if $p$ is algebraic$.$

$(d)$ $\mathbb Q + p$ is a proper subset of $\mathbb {R-Q}.$

My intuition says $(d)$ is the correct option though I may be wrong. But I can't figure out why is it correct if it really so. Would anyone please give me some hint to proceed?

Thank you in advance.

Answer 1

Note that $\mathbb{Q}$ and $\mathbb{Q}+p$ are countable, this immediately kills the first three options.

Answer 2

Yes ,(d) is correct since Q+p is a countable set and the Irrationals are uncountable ,