Statements:
A: $\forall x\in\mathbb R\exists y\in\mathbb R:(y^2-2xy+x^2 - 2x + 2y \leq 0)$
B: $\exists y\in\mathbb R\forall x\in\mathbb R:(y^2-2xy+x^2 - 2x + 2y \leq 0)$
Prove that statement A is true and disprove statement B
Can someone please help me with this question? I'm having a lot of trouble with these type of questions.
On
Hint for Part A. The question asks you to show that for all $x$, we can find $y$ such that some expression involving $x$ and $y$ is sufficiently small. So, why not let $x$ denote a fixed but arbitrary constant, and use basic calculus to find the $y$ that minimizes that expression. You'll get an expression $y(x)$ dependent on $x$ that makes the expression as small as possible. Then, just note that to prove
$$∀x∈\mathbb{R}∃y∈\mathbb{R}:P(x,y)$$
it suffices to prove
$$\forall x \in \mathbb{R} : P(x,y(x)).$$
Hint: you are dealing here with $\left(y-x+2\right)\left(y-x\right)$. Can you find for an arbitrary $x$ some $y$ that ensures this expression to be not positive? If so then you solved the first case. Can you prove that for every $y$ some $x$ can be found such that the expression is positive? Then you solved the second case.