You got your contradiction. Did you not?
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
Definitely not ,but by assuming the negated conclusion you ended up with:
But x+y+z ,hence
But we know that it is not true that ,hence .
Can you see now the contradiction??
Now you have to wander ,how the contradiction between two statements i.e ( p and notp ,in our case p= ) lead us to the desired result ,which in our case is:
You're going to look for a contradiction, once you have assumed the conclusion to be false.
But by falsifying the conclusion you can certainly arrive (as you did) at: x < 4, and y < 4, and z < 5
From this result you can certainly infer that x+y+z <= 10.
But, that is a blatant contradition of your hypothesis (i.e., that x + y + z >= 11).
Therefore, your assumption that the conclusion is false, is itself in fact false.
At this point (given that "reduction to absurdity" is an accepted method of proof) you're basically done, except for tying up some loose ends.