Explain the mistake in the following “proof.”
Theorem: The function defined by the formula for each integer n, is one-to-one.
“Proof: Suppose any integer n is given. Then by definition of f, there is only one possible value for namely, Hence f is one-to-one.”
To find out the mistake in the given proof.
Theorem: The function defined by the formula for all integers , is one-to-one.
Proof: Suppose any integer is given. Then by definition of , there is only one possible value for , namely . Hence is one-to-one.
The mistake in the proof is that how it can be said that “there is only one possible value for , namely ” without even proving it. Rather proof should have gone in the following way -
Let us suppose and assume that and now prove
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