Writing logic statements using quantifier

Question

I was given two predicates $text{Prime}(x)$ and $text{Even}(x)$ and is required to write the following statements:

For every odd natural number there is a different natural number such that their sum is even.

My attempt: $(forall x):(x in mathbb{N} wedge neg text{Even}(x) to (exists y):(x neq y wedge text{Even}(x+y))).$

and

The sum of any two prime numbers except the prime number $2$ is even.

My attempt: $(forall x,y):(x neq 2 wedge yneq 2 wedge text{Prime}(x,y) to text{Even}(x+y)).$

Is my attempt correct? And Am I allowed to write $text{Prime}(x,y)$ or should I write $(text{Prime}(x) wedge text{Prime}(y))?$

Air Mike · Answer

In the first statement, you wrote $x neq y wedge text{Prime}(x+y)$ but it should be $x neq y wedge text{Even}(x+y).$
Avoid writing $text{Prime}(x,y)$ because the predicate $text{Prime}$ only takes one argument (i.e., $text{Prime}(x)$.)
You could also have written the following.

$(forall x in mathbb{N})(exists y in mathbb{N}):[neg text{Even}(x) to (x neq y wedge text{Even}(x+y))].$

$(forall x in mathbb{N})(forall y in mathbb{N}):[(text{Prime}(x) wedge text{Prime}(y) wedge x neq 2 wedge y neq 2) to text{Even}(x+y)].$

Writing logic statements using quantifier

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