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What is the problem in this Proof

Amer

Active member
Mar 1, 2012
275
In your point of view what is the problem in this Proof
Claim any two natural a,b are equal
By induction
Let m= max{a,b}
if m=1 then a=b=1 since a,b natural
suppose it is hold for m=k
if
max{a,b} = k then a=b
test if
max{a,b} = k+1 , sub 1
max{a-1,b-1} = k which is the previous so a-1 = b-1 , a=b

I saw it in facebook
 

Bacterius

Well-known member
MHB Math Helper
Jan 26, 2012
644
In your point of view what is the problem in this Proof
Claim any two natural a,b are equal
By induction
Let m= max{a,b}
if m=1 then a=b=1 since a,b natural
suppose it is hold for m=k
if
max{a,b} = k then a=b
test if
max{a,b} = k+1 , sub 1
max{a-1,b-1} = k which is the previous so a-1 = b-1 , a=b

I saw it in facebook
[JUSTIFY]I saw it on math.stackexchange too. The problem is that $\max{(a, b)} = k ~ \implies ~ a = b$ is clearly wrong, and the proof was designed to hide this fact. For instance, $\max{(5, 7)} = 7$, but last time I checked we had $5 \ne 7$.

In essence, this "proof" doesn't show that all natural numbers are equal, it shows that any two equal natural numbers are equal ;)[/JUSTIFY]
 

Evgeny.Makarov

Well-known member
MHB Math Scholar
Jan 30, 2012
2,502
The problem is that $\max{(a, b)} = k ~ \implies ~ a = b$ is clearly wrong, and the proof was designed to hide this fact. For instance, $\max{(5, 7)} = 7$, but last time I checked we had $5 \ne 7$.
But this does not explain which proof step in particular is wrong. Of course the implication max(a, b) = k ⇒ a = b is false, just like the original claim that a = b for all a, b. But the proof claims to show just that, and the question is where the mistake in the proof is located.
 

caffeinemachine

Well-known member
MHB Math Scholar
Mar 10, 2012
834
In your point of view what is the problem in this Proof
Claim any two natural a,b are equal
By induction
Let m= max{a,b}
if m=1 then a=b=1 since a,b natural
suppose it is hold for m=k
if
max{a,b} = k then a=b
test if
max{a,b} = k+1 , sub 1
max{a-1,b-1} = k which is the previous so a-1 = b-1 , a=b

I saw it in facebook
The proof goes wrong in the last two sentences.

We have max{a,b}=k+1

Now we have max{a-1,b-1}=k.

We now want to apply the induction hypothesis here to have a-1=b-1 and thus a=b. But we can't do this. This is because we are not sure if a-1 and b-1 are natural numbers. We can very well have one of a-1 and b-1 equal to 0.

So this is the problem in the proof.