Writing logic statements using quantifier

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

1. 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

1. 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 $$(text{Prime}(x) wedge text{Prime}(y))?$$

Mathematics Asked by starry on December 29, 2020

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.

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

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

Answered by Air Mike on December 29, 2020

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