How do we symbolize the following sentence using logic?

Question

The square-root of 5 is irrational. [ 5: the number five; x · y: the product of x and y; I(x): x is a nonzero integer]

I tried to form some logical equivalences and identities, but they seemed to be incorrect.

Answer 1

We can also express your sentence as "There is no rational number, whose square is five". This yields the following expression: $$\neg\exists a, b \in \mathbb Z: I(b) \land \left(\frac ab\right)^2 = 5$$ Rearranging a bit to use only your terms, we have $$\neg\exists a, b: (I(a) \lor a = 0) \land I(b) \land a\cdot a = 5\cdot b\cdot b$$ Although this does not allow negative rational Numbers since you didn't give a symbol for that. Of course, since technically speaking the square root of a number always has to be positive, the expression remains correct.

Answer 2

Not sure quite what you're after, but one symbolization is:

$$\neg \exists n \in \mathbb Q : n\times n = 5$$

Read aloud, this is: "There does not exist a number $n$ in $Q$ (rational numbers) such that $n$ times $n$ equals $5$."