Question

Use proof by contradiction to veriify
if x^2-3x+2 < 0 then 1 9 years ago

Answer 1

@ganeshie8

Answer 2

Any proof by contradiction involves 2 steps :

1) Start by assuming the opposite of what you want to prove is true
2) Look for a contradiction

Answer 3

1) If x^2 - 3x+2< 0 then x<1 or x>2
Now what to do??

Answer 4

assume x^2 - 3x + 2 >= 0

is a suggestion

Answer 5

I dont follow..

Answer 6

oh I see what you are doing. That may be a good way to proceed

Answer 7

plug in value less than 1 and evaluate

Answer 8

But is this proof by contradiction? It is just a counterexample..

Answer 9

yea I'm very rusty at these...

Answer 10

-_-

Answer 11

Step 2 :

x^2 - 3x+2< 0
(x-1)(x-2) < 0

Case1 : x-1 < 0 and x-2 > 0
or
Case2 : x-1 > 0 and x-2 < 0

Answer 12

Wait a minute How is Case 1 possible?? Isn't the case 2 only valid ?

Answer 13

Case1 is impossible
Case2 yields the solution 1<x<2 which is a contradiction because we have assumed x<1 or x>2.

Answer 14

Why are we taking even Case 1 into account??

Answer 15

How do you know Case1 is impossible with out even looking at it ?

Answer 16

You should at least state that Case1 is impossible. Giving a reason for why you think it is impossible is much better.

Answer 17

Well because my teacher taught that if the a quadratic inequality is smaller than 0

Answer 18

that's pretty clever

Answer 19

Thus according to the above diagram, we dont even consider taking the Case 1

Answer 20

I am saying you should state exactly that in your proof.

Answer 21

Don't leave it to the imagination of the reader of your proof. You may simply say that Case1 doesn't give any solutions.

Answer 22

Okay I get that but I am asking why are we taking the Case 1 into account even though it will yield wrong results?

Answer 23

So its like saying
x+1 = 2
Case 1 x= 0 impossible
Case 2 x= 1 possible
Case 3 x=2 impossible

Answer 24

because we didn't know upfront that it would give wrong results

Answer 25

Well i have been taught that if inequality is <0 then the lower portion is taken and if >0 then the upper portion

Answer 26

Good for you.

Answer 27

So is that thing not universal?

Answer 28

I suggest you scroll up and read my reply for Step2 once again

Answer 29

Yeah you said it to state Both cases so as to not to leave it to the imagination of the reader

Answer 30

Yes, and ?

Answer 31

And I am saying if we can directly state that inequalities of the result by observing the sign in the given equality (if it is a universal thing) then why should we provide the Case 1 to the reader? (Is it because we are assuming he dont know?)

Answer 32

Depends on the reader. I wouldn't leave Case1 even if my reader has a phd in math as the proof wont be complete w/o that.

Answer 33

Okay thanks one more thing

Answer 34

Sure, ask