Is this true $\exists x\forall y (x + y > 8)$

Question

$U = \begin{Bmatrix} 1,2,3,4,5,6,7,8 \end{Bmatrix}$

Is this true? $\exists x\forall y (x + y > 8)$

My guess is yes? My thinking is that yes, there is atleast one x, 8, that makes every other element added bigger than 8? What do you guys think?

Answer 1

It is true: take $x=8$, then we have $x+y>8$ for all $y \in U.$

Answer 2

Yes, you pick $x=8$. And then: $$\text{y=1: }8+1>8$$ $$\text{y=2: }8+2>8$$ $$\text{y=3: }8+3>8$$ $$\text{y=4: }8+4>8$$ $$\text{y=5: }8+5>8$$ $$\text{y=6: }8+6>8$$ $$\text{y=7: }8+7>8$$ $$\text{y=8: }8+8>8$$ So the answer is yes.