Is S^{c} the key difference in this logical equivalence?

  • Thread starter Thread starter Horse
  • Start date Start date
  • Tags Tags
    Equivalent Logic
Click For Summary
SUMMARY

The discussion centers on the logical equivalence involving the set complement S^{c}. The original statement, ( ∀x ∈ S ∀y P(x) ) <=> ¬( ∃x ∈ S^{c} ∃y ¬P(x) ), is confirmed to be equivalent to the modified statement ( ∀x ∈ S ∀y P(x) ) <=> ¬( ∃x ∈ S ∃y ¬P(x) ). Participants agree that the key difference lies in the presence of S^{c}, which is crucial for understanding the logical relationship between the two expressions.

PREREQUISITES
  • Understanding of logical quantifiers: universal ( ∀ ) and existential ( ∃ )
  • Familiarity with set theory concepts, particularly set complements
  • Knowledge of logical equivalence and negation in mathematical logic
  • Basic proficiency in symbolic notation used in mathematical proofs
NEXT STEPS
  • Study the implications of set complements in logical expressions
  • Explore the principles of logical equivalence in mathematical proofs
  • Learn about the role of quantifiers in predicate logic
  • Investigate common logical fallacies related to negation and equivalence
USEFUL FOR

Mathematicians, computer scientists, and students of logic who seek to deepen their understanding of logical equivalences and set theory concepts.

Horse
Messages
34
Reaction score
0
Is it equivalent?
[tex] ( \forall x \in S \forall y P(x) ) <=> \neg ( \exists x \in S^{c} \exists y \neg P(x) )[/tex]

Attempt at solution
I think it should be
[tex] ( \forall x \in S \forall y P(x) ) <=> \neg ( \exists x \in S \exists y \neg P(x) )[/tex]
The diiference to the above statement is [tex]S^{c}[/tex].
 
Last edited:
Physics news on Phys.org


Yes, your statement is correct.
 


HallsofIvy said:
Yes, your statement is correct.

Please, can you clarify whether it is the first statement or my attempt at solution?
 

Similar threads

Writing some ZF axioms with FOL symbols
  • · Replies 2 ·
Replies
2
Views
2K
How to Interpret the Basic Logic Problem with Predicate P(x,y,z)
  • · Replies 2 ·
Replies
2
Views
2K
Prove ##(a+b) + c = a + (b+c)## using Peano postulates
  • · Replies 9 ·
Replies
9
Views
2K
Prove ##(a\cdot b)\cdot c =a\cdot (b \cdot c)## using Peano postulates
  • · Replies 1 ·
Replies
1
Views
1K
Undergrad Formal derivation of statement from Peano Arithmetic system
  • · Replies 10 ·
Replies
10
Views
2K
Prove ##(a+b)\cdot c=a\cdot c+b\cdot c## using Peano postulates
  • · Replies 3 ·
Replies
3
Views
2K
Expressing the existence of irrational numbers
  • · Replies 7 ·
Replies
7
Views
2K
Conversion of statements to math
  • · Replies 3 ·
Replies
3
Views
2K
Interpreting a statement in first order logic
  • · Replies 3 ·
Replies
3
Views
1K
Translating Sentences into Quantified Propositions: Practice Problems
  • · Replies 2 ·
Replies
2
Views
1K