Find the error in the \"proof\" of the following assertion. Any set of n element

Question

Find the error in the "proof" of the following assertion.

Any set of n elements has the property that all n elements are identical.

PROOF: By induction on n.

BASIS: n = 1 The set has one element a, and clearly a = a.

INDUCTION: Assume true for all sets of up to n - 1 elements and consider any subset of {a1, a2, ..., an}. If we form the subset {a1, a2, ..., an-1}, then by assumption a1 = a2 = ... = an-1. Also form the set {a2, a3, ..., an}. Collecting results, we have that a1 = a2 = ... = an-1 = an.

Explanation / Answer

we considered a set with number of elements=1

let set s = {1}

now the subset with n-1 element would be an empty [set. s ={} ]

so we cannot base our assumption on an empty set . Hence assuming that it is true for all sets of up to n - 1 elements and consider any subset of {a1, a2, ..., an}. If we form the subset {a1, a2, ..., an-1}, then by assumption a1 = a2 = ... = an-1 is itself false.

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