prove Here is a \"proof\" by induction that any two positive integers are equal

Question

prove

Here is a "proof" by induction that any two positive integers are equal (e.g., 5 = 10): First, a definition: if a and b are positive integers, define max(a,b) to be the larger of a and b if a b, and to be a if a = b. [For instance, max(3,5) = 5, max(3,3) = 3.] Let P(n) be the statement: "if a and b are positive integers such that max(a, b) = n, then a = b." We prove P(n) true for all n 1 by induction. [As a consequence if a, b are any two positive integers, then a = b, since P(n) is true where n=max(a,b).] First, P(1) is true, since if max(a,b) = 1 then a and b must both be equal to 1. Now assume P(n) true. Let a,b be positive such that max(a,b) = n+ 1. Then max(a-1,b-1) =n. As we are assuming P(n), this implies that a-1=b-1, hence a=b. Therefore, P(n + 1) is true. By induction, P(n) is true for all n. There must be something wrong with this "proof. Can You error?

Explanation / Answer

yes there is mistake in the proof,

from P(n) is true, you find you found out that a=b

but from P(n)=max(a,b)=n

this implies, a=b=n

so for P(n+1)=max(a,b)=n+1, is wrong because max(a,b)=a=b=n+1

which is contradicting our previous assumption that a=b=n.

so this is the mistake in the proof

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