Show that the logic $ \ \forall x \exists y \ P(x,y) \ $ is not logically equivalent to $ \ \exists x \forall y \ P(x,y) \ $ in the **domain** $ \ \{-1,0,1 \} \ $ by giving a **counter example** .

**Answer:**

Let 

$ x=\{0,1 \} \\ y=\{-1,0 ,1 \} \ $

Let $ P(x,y) \ $ be the property such that $ \ x \geq y \ $ 

Then , $ \forall x \exists y \ P(x,y) \ $ is true. 

But, $ \ \exists x \forall y \ P(x,y) \ $ is False. Because for $ x=0 \ $ , we have $ \ 0 \ngeqslant y=1 \ $.

**I need confirmation of my work.**