I'm studying discrete mathematics and I'm faced with a problem of converting a few descriptive mathematical statements into logical notation. Any help would be appreciated. Thank you.
a) Any divisor of an integer is less than or equal to itself.
b) Any integer greater than $1$ has at least one prime divisor.
I'm supposed to solve it using the "usual arithmetic operators: $+, -, *, /, =, \ne, <, >,\le$, and $\ge$".
I have attempted part (a):
$$ (n\in \mathbb{Z} \wedge x \mid n)\to (x \le n) $$
Here $x$ is any arbitrary number that divides an integer. Is it right?
Mostly. You need quantifiers for "any".
$$\forall x\; \forall n :\Bigl( x,n\in \mathbb Z\wedge (x\mid n)\implies(x\leq n)\Bigr)$$
Sometimes written as : $\forall x \in \mathbb Z, \forall n \in \mathbb Z: (x\mid n)\implies (x\leq n)$
Likewise, for part b you'll need to identify that there exists a prime, somewhere in the statement. $$\exists m: m\in\mathbb P$$
Can you do it now?