What is wrong with the following argument, which allegedly shows that any two positive integers are equal?
We use induction on n to “prove” that if a and b are positive integers and n = max {a, b}, then a = b = 1.
Basis Step (n = 1) If a and b are positive integers and 1 = max{a, b}, we must have a = b = 1.
Inductive Step Assume that if a´ and b´ are positive integers and n= max{a´, b´}, then a' = b'. Suppose that a and b are positive integers and that n + 1 = max{a, b}.
Now n = max{a – 1, b – 1). By the inductive hypothesis, a – 1 =b – 1. Therefore, a = b.
Since we have verified the Basis step and the Inductive by Step, by the Principle of Mathematical Induction, any two positive integers are equal!
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