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

In summary, two statements are considered equivalent in logic if they have the same truth value in all possible cases. This can be determined by creating a truth table for both statements or by using logic laws and rules. Even if two statements have different structures, they can still be equivalent as long as their truth values are the same. The concept of equivalence is important in logic as it allows us to simplify and clarify statements, make logical arguments more concise, and understand the relationships between different statements.
  • #1
Horse
35
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:
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  • #2


Yes, your statement is correct.
 
  • #3


HallsofIvy said:
Yes, your statement is correct.

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

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

What does it mean for two statements to be equivalent in logic?

Two statements are considered equivalent in logic if they have the same truth value in all possible cases. In other words, if the truth value of one statement is true, then the truth value of the other statement is also true, and if the truth value of one statement is false, then the truth value of the other statement is also false.

How can I determine if two statements are equivalent?

One way to determine if two statements are equivalent is by creating a truth table for both statements. If the truth values of the two statements are the same for all possible combinations of truth values of their component propositions, then the statements are equivalent.

Are there any shortcuts for determining equivalence in logic?

Yes, there are several laws and rules in logic that can be used to determine equivalence without having to create a truth table. Some of these include the distributive law, De Morgan's laws, and the law of double negation.

Can two statements be equivalent even if they have different structures?

Yes, two statements can be equivalent even if they have different structures. The structure of a statement refers to the way it is composed, such as the use of logical operators like "and" and "or". As long as the truth values of the statements are the same for all possible cases, they are considered equivalent.

How is the concept of equivalence used in logic?

The concept of equivalence is used in logic to simplify and clarify statements. By determining equivalence, we can reduce the number of statements we need to consider and make logical arguments more concise. It also allows us to see connections between seemingly different statements and helps us understand the relationships between them.

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