Que. Consider domain is the set of all people in the world.
$F(x,y) =x \text{ is the friend of y}.$
Represent each of the following sentences using first-order logic statements
$1.$ Every person has $at most \ 2$ friends.
$2.$ Every person has $exactly \ 2$ friends.
$3.$ Every person has $at least \ 2$ friends.
My attempt –
$1. \forall x \exists y_1\exists y_2(F(x,y_1) \wedge F(x,y_2) \wedge \forall z(F(x,z) \implies ((z= y_1) \vee (z= y_2))))$
$2. \forall x \exists y_1\exists y_2(F(x,y_1) \wedge F(x,y_2) \wedge (y_1 \neq y_2) \wedge \forall z(F(x,z) \implies ((z= y_1) \vee (z= y_2))))$
$3. \forall x \exists y_1\exists y_2(F(x,y_1) \wedge F(x,y_2) \wedge (y_1 \neq y_2))$
Please verify.