What are the negations of complex statements in statement logic?

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The discussion focuses on negating complex statements in statement logic, specifically addressing two examples: (i) ∀x∈R ∃y∈R such that x+y=0 and (ii) a scenario involving individuals finding red apples in bags. The correct negation of statement (ii) is "There exists one of us who hasn't found at least one red apple in at least one bag." The participants emphasize the importance of understanding the logical structure, including the use of quantifiers such as "for all" (∀) and "there exists" (∃).

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1. Negate the following statements

(i): [tex]\forall[/tex]x[tex]\in[/tex]R [tex]\exists[/tex]y[tex]\in[/tex]R 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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