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?

The form of the statement that you need to prove is "if A then B". When you speak about the "or" in your statement, you are discussing whether B is true. But recall that "if A then B" does NOT imply that "if B then A". Let's apply that to your example...
Let $x,y\in\mathbb{R},\:x=12,\:y=55$.
Then $x+y=67\lt100$.
Now let A = "$x+y\geq100$" and B = "$x\geq50$ or $y\geq50$".
Given our values for $x$ and $y$, we have A is false and B is true.
So we have "if A then B" is true and "if B then A" is false.
It's still up to you to prove the statement, but we've shown why your example is not really a counter-example.
