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yxgao
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What is the negation of the statement "For each s in R, there exists an r in R such that if f(r) >0, then g(s) >0."
Originally posted by yxgao
matt grime's answer is correct.
thanks :)
DeadWolfe, all you did was restate the question.
Originally posted by yxgao
What is the negation of the statement "For each s in R, there exists an r in R such that if f(r) >0, then g(s) >0."
StarThrower said:Original Statement:
[tex] [\forall s \in R[\exists r \in R(f(r) > 0 \supset g(s) > 0)]][/tex]
Prefix Of Original statement by the logical operator NOT:
[tex]\neg [\forall s \in R[\exists r \in R(f(r) > 0 \supset g(s) > 0)]][/tex]
DISCUSSION
As you can see, all quantifiers used in the original statement refer to the same domain of discourse, namely the set of real numbers. In the orginal formulation of the statement, the domain of discourse is made explicit (we can see the script R symbol denoting the set of real numbers). But, when all the quantifiers used in a statement refer to the same domain of discourse, there is a shorter way to formulate the statement, but it must be made clear at the outset, that all the quantifiers used in the statement refer to the set of real numbers. The only time we must specify what set a quantifier refers to, is when we are talking about multiple domains of discourse. Since there is only one domain of discourse here, the orginal statement is more succinctly formulate as follows:
Let the domain of discourse be the set of real numbers.
Original statement(domain of discourse implicit, instead of explicit)
[tex]\neg [\forall s [\exists r \( f(r) > 0 \supset g(s) > 0]][/tex]
Now, we can pass the negation through the quantifiers (reversing them), and it operates on the conditional:
[tex]\exists s \forall r \[ \neg [f(r) > 0 \supset g(s) > 0 ][/tex]
if X then Y = not (X and not Y), hence
not (if X then Y) = not (not(X and not Y)) = X and not Y
So, we can formulate the negation of the original statement as follows:
[tex]\exists s \forall r \ [ f(r) > 0 and \neg g(s) > 0 ][/tex]
All the formulations are semantically equivalent, in other words, they all mean the same thing.
Translation: Let the domain of discourse be the set of real numbers. There is at least one s such that for any r, f(r) is greater than zero and not ( g(s)>0 ).
(by saying the domain of discourse is the set of real numbers, it is known by he who formulated the statement, that s,r are elements of the set of real numbers.
First order logic examples:
There is at least one dog, who is older than any cat.
In the previous example, there are TWO domains of discourse, instead of one.
Let D denote the set of dogs, and let C denote the set of cats.
We can formulate the statement as follows:
[tex] \exists d \in D\forall c \in C [d-is older than-c] [/tex]
Let us suppose that the previous statement is true.
Now, consider the meaning of the previous statement. There is at least one dog, who is older than any cat. So, suppose that wonderdog is such a dog, then it follows that:
[tex] \forall c \in C [Wonderdog-is older than-c] [/tex]
Suppose that Hairball is an element of the set of cats. If the previous statement is true, then the following statement is true:
Wonderdog is older than Hairball.
My point is this: That when we are talking about multiple domains of discourse, we need to write out which set a quantifier refers to explicitely, but when all the quantifiers refer to the same single set, we can use the implicit notation, which is shorter.
The negation of a statement is the opposite of the original statement. It is a statement that is true if the original statement is false, and false if the original statement is true.
In logic, the negation of a statement is typically represented by the symbol "~". This symbol is placed in front of the original statement to indicate that it is being negated.
Yes, the negation of a statement can be more than one sentence. It can also be represented by multiple symbols, such as "~" for "not" and "&" for "and". However, it is important to make sure that the negation accurately reflects the original statement.
The negation of a statement flips its truth value. This means that if the original statement is true, the negation will be false, and if the original statement is false, the negation will be true.
The negation of a statement can be proven to be true or false through logical reasoning and evidence. However, it is important to note that the negation of a statement is not always true, as it depends on the truth value of the original statement.