I have the following questions:
Write the following statements in more abbreviated form, using quantifiers. Here the short phrases “is prime” and “is a line” are allowed, and the symbol $\Pi$ may be used for “the plane.”
(a) 17 is not the largest prime number.
My Attempt: ($\exists x \in \Bbb P ) (x > 17 )$
(b) There is no largest prime number.
My Attempt: $(\forall x \in \Bbb P )(\exists y \in \Bbb P ) (x < y)$
(c) Every real number has a fifth root.
My Attempt: $(\forall x \in \Bbb R )(\exists y \in \Bbb R ) (y^5 = x )$
(d) Every pair of distinct points in the plane lies on a unique line.
I am still unsure about my attempts. Could some confirm whether my attempts are correct or not and point me in the right direction when it comes to part (d).
You've done very well on $(a) - (c)$. Your translations on those are entirely correct.
For (d): "Every pair of distinct points in the plane lies on a unique line."
$$(\forall x)\,(\forall y)\,[(x \in \Pi \,\land\, y \in \Pi\, \land\, x \ne y) \rightarrow (\exists ! \,\mathcal l)(Line(\mathcal l) \land x \in l \land y \in \mathcal l)]$$
Here, $\exists !$ denotes "There exists a unique".
If you haven't yet encountered this quantifier, we can still translate $(d)$ without it, though the statement becomes more elaborate:
$$(\forall x)\,(\forall y)\,[(x \in \Pi \,\land\, y \in \Pi\, \land\, x \ne y) \rightarrow (\exists \,\mathcal l)(Line(\mathcal l) \land x \in l \land y \in \mathcal l\land (\forall \mathcal l')(\mathcal l' \neq \mathcal l \rightarrow (x \notin \mathcal l' \lor y \notin \mathcal l'))]$$