negation of statement with "at least"

Question

The negation of the following sentence:

At least one girl has bought her friend a book, which she hasn't read herself.

I have tried to write it with logical symbols

p = at least one girl has bought her friend a book

q = she has not read the book herself

so would the negation be $\lnot( p \land q)$, and this would be $\lnot p \lor \lnot q$?

but how could I say it in english?

would the negation of "at least" be "All girls....?"

Answer 1

To answer the first part of your question, yes. $\lnot(p \land q) = \lnot p \lor \lnot q$.

To answer the second, think when "at least one girl has blah" is true. Is it true when $0$ girls have blah? Is it true when $1$ girl has blah? Is it true when $2$ girls have blah?

Since "at least one girl has blah" is true for $\geq 1$ girls, the only time it fails is when $0$ girls have blah. So that's the negation.

As mentioned in the comments, you may notice that "at least one" is the same thing as "there exists". So if you know that $\lnot \exists x \varphi = \forall x \lnot \varphi$, you can do this slightly quicker.

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I hope this helps ^_^

Answer 2

First part of your questions is by De Morgan’s law. And second part is also correct. “At least one girl did something” is equivalent to saying that “there exists a girl who did something”. The negation of “there exists” is “for all”.