Translation of : "the sum of two prime numbers each greater than 3 is even" in First Order Logic

Question

I have used these symboles :

$Z(x) : x$ is a prime number
$f(x,y) : x + y$
$L(x,y) : x\ge y$
$t: 3$
$E(x) : x$ is even

$$∀x∀y(Z(x)∧Z(y)∧L(x,t)∧L(y,t))\rightarrow E(f(x,y)))$$

Is this a correct translation ?

Answer 1

Your title says "the sum of two prime numbers each greater than $3$..." but your propositional function $L$ states $x$ is greater than or equal to $y$. I will assume the latter is what you intended.

Also, you're missing a left paranthesis "$($." To help with readability, I would actually drop some parantheses and take advantage of operator precedence. $\wedge$ comes before $\to$.

In addition, you can drop the definiton of $t$ and insert $3$ directly into propositional functions $F$ and $L$. This will make everything more compact.

As a final note, you should always specify the doman of your variables. In other words, follow up your proposition with a brief statement about the domains of $x$ and $y$

For instance...

$Z(x) : x$ is a prime number
$f(x,y) : x + y$
$L(x) : x \ge 3$
$E(x) : x$ is even

$$ ∀x∀y ( Z(x)∧Z(y)∧L(x)∧L(y) \to E(f(x,y)) ) $$

where $ x,y\in \mathbb{Z}$