I am quite lost on proving or disproving these type of statements, if anyone can guide me for a way to approach this problem that would be great.
Determine whether the statement is true or false, and prove or disprove it: $(\forall x \in \mathbb{R})(\exists y \in \mathbb{R})(\forall z \in \mathbb{R})[xy = xz]$
This is how I went by trying to see if its true or false:
I picked an x = 2, and y = 3, and a z = 3. Then I got $(2*3)=(2*3) \rightarrow 6 = 6$, which is true?
Best Answer
I am going to show that this is false. Take the negation of the original statement $(0)$: $$\exists x\in\Bbb R\forall y\in\Bbb R\exists z\in\Bbb R xy\ne xz\tag1$$ Now let $x=1$. $$\forall y\in\Bbb R\exists z\in\Bbb R y\ne z\tag2$$ Clearly we can take $z=y+1$ to make $y\ne z$. $(2)$ is thus true, and since $x=1$ is the element for which this was proved true, $(1)$ is true and $(0)$ is false.