I am reading "Set Theory: A First Course" by Daniel Cunningham.
There is the following exercise in this book (Exercise 1.4.6 on p.23):
Let $\phi(x)$ be a formula. What does $\forall z\forall y((\phi(x)\land\phi(y))\to z=y)$ assert?
My 1st answer is here:
If $z\neq y$, then $\phi(x)$ or $\phi(y)$ doesn't hold.
My 2nd answer is here:
$\phi(x)$ doesn't hold or if $z\neq y$, then $\phi(y)$ doesn't hold.
My 3rd answer is here:
$\phi(x)$ doesn't hold or if $\phi(y)$ holds, then $z=y$.
Are my answers ok?
By the way, there is the following exercise in this book (Exercise 1.4.1 on p.23):
What does the formula $\exists x\forall y(y\notin x)$ say in English?
I guess the correct answer is $x=\emptyset$.
Is the following answer ok or not? :
There exists a set $x$ such that $y\notin x$ for all $y$.