Need Help in Conversion to English

Question

I have to convert the given predicates to English sentences. I have tried from my side but I am very confused (as there can be many forms to represent a statement and also that my English statement might be wrong). Here are the predicates and English statements. Please tell me if there can be a better answer for any of these?

1. $\forall x\exists y R(x,y)$

   For every $x$ there is at least one $y$ such that $R(x,y)$

2. $\exists x\forall y R(x,y)$

   For every $x$ there is a single $y$ such that $R(x,y)$

   EDIT:

   There is an x $x$ for which there is a single $y$ such that $R(x,y)$

3. $\forall x(\neg Qx)$

   For every $x$ $Q(x)$ is false; or $Q(x)$ is false for every $x$

4. $\exists y(\neg P(y))$

   There is at least one $y$ for which $P(y)$ is false.

Please help me out. Thanks.

Answer 1

You have only mistake in number 2 as Arturo has pointed. The right expression is: There exists at least one $x$ such that for all $y$ the formula $R(x,y)$ is true. What you wrote is encoded as $\forall x\exists yR(x,y)$.