Expressing statements using quantifiers

438 Views Asked by At

I would like to confirm my answers for these questions before carrying out with more studying--since there are no answers, and I'm worried I will be practicing incorrectly.

Let F(x)="x is friendly", T(x)="x is tall" and A(x)="x is angry" note: ~ = negation

Ex 1. Some people are not angry

My answer: ∃x~A(x)

Ex 2. All tall people are friendly

My answer: ∀xT(x) -> F(x)

Ex 3. No friendly people are angry

My answer: ∀xF(x) -> ~ A(x)

I'm feel confident the answers are correct, but just in case.

1

There are 1 best solutions below

0
On

In the second and third answers the quantifier is intended to apply to the entire expression that follows it, which therefore needs to be in parentheses: $$\forall x\big(T(x)\to F(x)\big)$$ and $$\forall x\big(F(x)\to\neg A(x)\big)\;.$$ $\forall xT(x)\to F(x)$ without the extra parentheses means $\big(\forall T(x)\big)\to F(x)$, which doesn’t actually say anything, since the $x$ in $F(x)$ is a so-called free variable: it’s an unquantified placeholder. The closest you can come to an English translation is If everyone is tall, then x is friendly, but $x$ doesn’t actually refer to anyone in particular.

In other words, quantifiers behave rather like the negative sign in arithmetic and algebra: if you want the negative of $a+b$, you need to use parentheses and write $-(a+b)$, because $-a+b$ means $(-a)+b$.

Because of the negation I prefer to use the extra parentheses even in your first expression, writing $\exists x\big(\neg A(x)\big)$, but some people are comfortable with your version.