What are the negations of complex statements in statement logic?

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Negating complex statements in logic involves understanding the quantifiers and their relationships. For the first statement, "For all x in R, there exists a y in R such that x + y = 0," the negation is "There exists an x in R such that for all y in R, x + y ≠ 0." The second statement, which asserts that at least one person found at least one red apple in at least one bag, negates to "No one found a red apple in any bag." The discussion highlights confusion around negating multi-part statements and seeks clarification on the process. Understanding the structure of these statements is crucial for accurately forming their negations.
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1. Negate the following statements

(i): \forallx\inR \existsy\inR such that x+y=0

(ii): introduction: Each of us got, let's say, 20 bags of green apples.
Actual Statement: At least one of us (each) found at least one red apple in at least one bag (each). (Each person and each one of it's bags are treated separately)

2. Homework Equations /3. The Attempt at a Solution
No idea. I'm completely lost here.

Please help, as I have no idea how to negate complex/multi-element statements.
 
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Welcome to PF!

test_notagain said:
1. Negate the following statements
…
At least one of us (each) found at least one red apple in at least one bag (each). (Each person and each one of it's bags are treated separately) …

Hi test_notagain! Welcome to PF! :smile:

(have an exists: ∃ and an in: ε and a for-all: ∀ :wink:)

Let's try (ii) first …

the opposite of something beginning "At least one of us has …" is "There exists one of us who hasn't …"

can you go on from there? :smile:
 


Thanx tiny-tim, but I still don't get it because there are 3 parts to statement (ii)... And what is the negation of the first one (i)?
 

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