# Translating "If there were no computers with antivirus every computer would work fine."

"If there were no computers with antivirus every computer would work fine".

Use: O(x) = x is a computer, A(x) = Computer x has an antivirus, F(x) = x works fine.

My take is $$∀x((O(x)∧¬A(x))→F(x)).$$ What I'm trying to say here is that if every computer doesn't have an antivirus, then it must work fine no matter what. But my professor tells me it's not like that.

• Maybe A(x) = Computer x has an antivirus Mar 27 at 8:38
• Your formula means: "All computers without antivirus work fine". This is not the same statement as the one given.
– hff1
Mar 27 at 8:43
• You can rephrase (b) as "at least one number is a divisor of 3 and if x and y are divisors of 3 then x equals y" Mar 27 at 9:40

My take is: $$∀\color\red x\Big((O\color\red x∧¬A\color\red x)→F\color\red x\Big)$$
The antecedent "there are no computers with antivirus" means $$\lnot\exists x\:(Ox\land Ax);$$ the consequent "every computer works fine" means $$\forall x\:(Ox\to Fx);$$ putting them together: $$\lnot\exists x\:(Ox\land Ax)\to\forall x\:(Ox\to Fx);$$ more clearly: $$\lnot\exists \color{violet}x\:(O\color{violet}x\land A\color{violet}x)\to\forall \color{brown}y\:(O\color{brown}y\to F\color{brown}y).$$
P.S. Whatever it is, do not make the common mistake of writing $$\forall x \,(Px\to Q)$$ as $$\forall x Px\to Q;$$ they are not logically equivalent!