I need help understanding contradiction. I know that you are supposed to negate the conclusion of the problem. But after that, I am unsure of what to do next. Here is a practice problem I have:
Prove the following statement by contradiction: “if integers x, y, z satisfy x + y + z >= 11, then either x >= 4, or y >= 4, or z >= 5.”
The hypothesis is: if integers x, y, z satisfy x + y + z >= 11
The conclusion is: then either x >= 4, or y >= 4, or z >= 5
The negated conclusion is: then x < 4, and y < 4, and z < 5
At this point I am unsure of what to do next. Am I supposed to find the numbers for x, y, z to satisfy the negated conclusion. If I use X = 3, y =3, and z = 4, I get 3 + 3 + 4 = 10. 10 < 11.
Did I just solve the problem because all I have to do is find one way to satisfy the negated conclusion which in turn proves the real conclusion? Or did I totally miss something?
Thanks in advance