What are the negations of complex statements in statement logic?

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In summary, the negation of (i) is: ∃x∈R ∀y∈R such that x+y ≠ 0. The negation of (ii) is: ∀x∈R ∃y∈R such that x+y ≠ 0.
  • #1
test_notagain
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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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  • #2
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:
 
  • #3


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)?
 

1. What is statement logic?

Statement logic, also known as propositional logic, is a branch of formal logic that studies the relationships between statements or propositions. It uses symbols and rules to determine the truth value of compound statements based on the truth values of its component statements.

2. What is negation in statement logic?

Negation is a logical operation that reverses the truth value of a statement. In statement logic, the negation of a statement is represented by the symbol ¬ or ~. For example, the negation of the statement "It is raining" would be "It is not raining."

3. What is the difference between negation and contradiction?

Negation is a logical operation that changes the truth value of a statement, while contradiction is a statement that is always false regardless of its truth value. In other words, negation is a logical operation, while contradiction is a type of statement.

4. How is negation used in everyday language?

Negation is commonly used in everyday language to express denial or disagreement. For example, saying "I am not hungry" is a negation of the statement "I am hungry." It can also be used to express the absence of something, such as "There is not a single cloud in the sky."

5. What are the basic rules of negation in statement logic?

The basic rules of negation in statement logic are:

  • The negation of a true statement is false, and the negation of a false statement is true.
  • The double negation of a statement is equivalent to the original statement.
  • The negation of a conjunction (AND) is equivalent to the disjunction (OR) of the negations of the individual statements.
  • The negation of a disjunction (OR) is equivalent to the conjunction (AND) of the negations of the individual statements.
  • The negation of an implication (IF-THEN) is equivalent to the conjunction (AND) of the original statement and the negation of the consequent.
  • The negation of a biconditional (IF AND ONLY IF) is equivalent to the disjunction (OR) of the conjunctions (AND) of the original statement and the negation of the consequent, and the negation of the antecedent and the original statement.

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