Logic (Negation of an expression)

Question

$\mathcal P(x,y)$ is a predicate meaning "$x$ is $y$'s brother"

Write the negation of the following assertion:

$$\forall x \forall y~(\mathcal P(x,y) \Rightarrow \mathcal P(y,x))$$

Above is the exercice.

The solution I got is

$$\exists x \exists y ~(\mathcal P(x,y) \land \lnot\mathcal P(y,x))$$

I am however not sure wether this is right and the professor did not provide us with any soltuions. Any help would be greatly appreciated!

Answer 1

Typographical errors aside, indeed the expressions are equivalent.   Be careful with bracket placement.

$$\neg \forall x\forall y ~(\mathcal P(x,y)\to\mathcal P(y,x)) ~\iff~ \exists x\exists y~(\mathcal P(x,y)\wedge \neg \mathcal P(y,x))$$

You have applied dual negation to the quantifiers, and negated the conditional, correctly.

Answer 2

Yes you are right.

The logic behind is that $A\Rightarrow B$ is equivalent with $\neg A\vee B$. Consequently $\neg(A\Rightarrow B)$ is equivalent with $\neg(\neg A\vee B)=A\wedge \neg B$.