What is the negation of this statement and which statement is true, R or ¬R?

  • MHB
  • Thread starter rayne1
  • Start date
In summary, Rayne's answer is that R = (∀x ∈ Z)(P(x) ⇒ ((∃y ∈ Z)(E(y) ∧ D(x, y)))) and ¬R = (∃x ∈ Z)(P(x) ∧ (∀y ∈ Z)((¬E(y)) ∨ (¬D(x, y)))).
  • #1
rayne1
32
0
Let P(x) be assertion “x is prime”, E(x) be “x is even”, and D(x, y) be “x divides y” (i.e., y/x is an integer). Consider the following statement:
R = (∀x ∈ Z)(P(x) ⇒ ((∃y ∈ Z)(E(y) ∧ D(x, y))))

Write the negation of R, and determine which statement is true, R or ¬R.

I tried, but I'm not sure if I got the correct answer:
¬R = (∃x ∈ Z)(P(x) ∧ (∀y ∈ Z)((¬E(y)) ∨ (¬D(x, y)))

It seems that ¬R is true.
 
Physics news on Phys.org
  • #2
rayne said:
Let P(x) be assertion “x is prime”, E(x) be “x is even”, and D(x, y) be “x divides y” (i.e., y/x is an integer). Consider the following statement:
R = (∀x ∈ Z)(P(x) ⇒ ((∃y ∈ Z)(E(y) ∧ D(x, y))))

Write the negation of R, and determine which statement is true, R or ¬R.

I tried, but I'm not sure if I got the correct answer:
¬R = (∃x ∈ Z)(P(x) ∧ (∀y ∈ Z)((¬E(y)) ∨ (¬D(x, y)))

It seems that ¬R is true.

Hi rayne!

Let's define:
Q(x) = (∃y ∈ Z)(E(y) ∧ D(x, y))

Then we have:
R = (∀x ∈ Z)(P(x) ⇒Q(x))

So:
¬R = (∃x ∈ Z)(¬(P(x) ⇒Q(x)))

Did you know that:
(P(x) ⇒Q(x)) = (P(x) ∨ ¬Q(x))
(Wondering)

So what does that make ¬(P(x) ⇒Q(x))?
 
  • #3
I like Serena said:
Hi rayne!

Let's define:
Q(x) = (∃y ∈ Z)(E(y) ∧ D(x, y))

Then we have:
R = (∀x ∈ Z)(P(x) ⇒Q(x))

So:
¬R = (∃x ∈ Z)(¬(P(x) ⇒Q(x)))

Did you know that:
(P(x) ⇒Q(x)) = (P(x) ∨ ¬Q(x))
(Wondering)

So what does that make ¬(P(x) ⇒Q(x))?
P(x)∧(¬Q(x))
 
  • #4
rayne said:
P(x)∧(¬Q(x))

How did you get that?
 
  • #5
I like Serena said:
How did you get that?
I know that they're logically equivalent
 
  • #6
rayne said:
I know that they're logically equivalent

So you're saying that:
¬(P(x) ⇒Q(x)) = (P(x)∧(¬Q(x)))

Let's take another look.
Substitute (P(x) ⇒Q(x)) = (P(x) ∨ ¬Q(x)) to get:
¬(P(x) ⇒Q(x)) = ¬(P(x) ∨ ¬Q(x))

... but wait! Isn't that the negation of what you are saying? :eek:
 
  • #7
I like Serena said:
So you're saying that:
¬(P(x) ⇒Q(x)) = (P(x)∧(¬Q(x)))

Let's take another look.
Substitute (P(x) ⇒Q(x)) = (P(x) ∨ ¬Q(x)) to get:
¬(P(x) ⇒Q(x)) = ¬(P(x) ∨ ¬Q(x))

... but wait! Isn't that the negation of what you are saying? :eek:

I guess so.
 
  • #8
I have to go right now, so can't reply in the next 24 hours.
 
  • #9
rayne said:
Let P(x) be assertion “x is prime”, E(x) be “x is even”, and D(x, y) be “x divides y” (i.e., y/x is an integer). Consider the following statement:
R = (∀x ∈ Z)(P(x) ⇒ ((∃y ∈ Z)(E(y) ∧ D(x, y))))

Write the negation of R, and determine which statement is true, R or ¬R.

I tried, but I'm not sure if I got the correct answer:
¬R = (∃x ∈ Z)(P(x) ∧ (∀y ∈ Z)((¬E(y)) ∨ (¬D(x, y)))

It seems that ¬R is true.
Rayne, you are right on both counts.

I like Serena said:
Did you know that:
(P(x) ⇒Q(x)) = (P(x) ∨ ¬Q(x))
No, (P(x) ⇒ Q(x)) ⇔ (¬P(x) ∨ Q(x)).
 
  • #10
Evgeny.Makarov said:
Rayne, you are right on both counts.

Let's read what R says:
For every x in Z, if x is prime, then there is an y such that y is even and x divides y.
I believe this is true. Pick y=2x.

No, (P(x) ⇒ Q(x)) ⇔ (¬P(x) ∨ Q(x)).

Ah. My bad.
 
  • #11
I like Serena said:
Let's read what R says:
For every x in Z, if x is prime, then there is an y such that y is even and x divides y.
I believe this is true. Pick y=2x.
Yes, of course. I must have read $D(x,y)$ the other way. The negation is still constructed correctly, I believe.

Well, this thread has some unfortunate responses, doesn't it?
 

FAQ: What is the negation of this statement and which statement is true, R or ¬R?

What is the definition of "negation of this statement"?

The negation of a statement is the opposite or contradictory of the original statement. It is denoted by the symbol "~" or "not". For example, the negation of the statement "It is sunny today" would be "It is not sunny today".

How is the negation of a statement expressed in logic and mathematics?

In logic and mathematics, the negation of a statement is typically expressed using symbols such as "~" or "not". For example, the statement "All dogs are mammals" can be negated as "~(All dogs are mammals)" or "It is not true that all dogs are mammals".

What is the purpose of using the negation of a statement in scientific research?

The negation of a statement is used in scientific research to test the validity of a hypothesis or to consider alternative explanations. By negating a statement, scientists can explore different perspectives and challenge their own assumptions, leading to a more thorough and accurate understanding of the subject being studied.

Can the negation of a statement ever be proven to be true?

No, the negation of a statement cannot be proven to be true. It can only be proven false by providing evidence that supports the original statement. This is because the negation of a statement is not a statement in itself, but rather the opposite or contradictory of a statement.

How does the use of double negation affect the meaning of a statement?

Double negation refers to the use of two negative words or symbols in a statement, which can cancel each other out. In logic and mathematics, it is generally accepted that double negation results in affirming the original statement. However, in everyday language, double negation can often lead to confusion and can change the intended meaning of a statement.

Similar threads

Replies
1
Views
798
Replies
1
Views
827
Replies
1
Views
1K
Replies
4
Views
1K
Replies
1
Views
1K
Replies
1
Views
2K
Replies
2
Views
1K
Back
Top