Prove that for all real numbers $x$ and $y$, if $x+y \geq 100$, then $x \geq 50$ or $y \geq 50$. I'm confused about the following question in my math textbook.
Prove that for all real numbers $x$ and $y$, if $x+y \geq 100$, then $x \geq 50$ or $y \geq 50.$
The or is what gets me. For or to be true don't we need only one of the statements in the operation to be true? Couldn't we have $x = 12, y = 55, x+y = 67$ is there something I'm missing here? Shouldn't it be an and instead of an or?

"Before taking this course, you must have passed either a course on omphalology or a course on exorcism of monsters."
The "or" above clearly means one or the other or both.
It's standard in mathematical reasoning to construe "or" that way.
