Consider $P=\forall x_1,x_2 \in \mathbb R [(\exists y\in \mathbb R [ (x_1,y)\in B \land (x_2,y)\in B ]) \implies (x_1=x_2)]$

Question

Let P be the following statement:

$\forall x_1,x_2 \in \mathbb R [(\exists y\in \mathbb R [ (x_1,y)\in B \land (x_2,y)\in B ]) \implies (x_1=x_2)]$

Let $B= \{(t,t^2) | t\in \mathbb R\}$

Does B satisfy P?

The answer says it does not, but wouldn't $y=0$ work? Then $0^2=0=x$

Answer 1

For the statement $\;P\;$ to be true for the $\;B\;$ you defined , the condition must be fulfilled always.

Well, now check that $\;(-1,1)\in B\;,\;\;(1,1)\in B\; $ ...