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## Homework Statement

Why is the negation of: Atleast one hole contains atleast n+1 pigeons where n is a natural number

There are less than n+1 pigeons per hole?

Wouldn't it be: There are less than one hole that contains less than n+1 pigeons?

Wouldn't the negation of there exists a hole that contains atleast n+1 pigeons yield

all holes contain less than n+1 pigeons?

Why is the existential quantifier assumed for the phrase: "Atleast one hole" in the first part of the sentence but not for the "atleast n+1 pigeons"?

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