How Effective Are These Logical Negations?

[Mr Davis 97]

Homework Statement

Homework Equations

The Attempt at a Solution

1) There exists a grey cat.
Negation: Every cat is not grey.

2) For all cats there exists an owner.
Negation: There exists a cat without an owner.

3) There exists a grey cat for all owners.
Negation: There exists an owner that can't own a grey cat.

4) Every fire engine is red and every ambulance is white.
Negation: There exists a fire engine that isn't red or there exists an ambulance that isn't white.

5) Some of the students in the class are not here today.
Negation: Every student is in class today.

6) Let $x,y,z \in \mathbb{Z}$. For all x there exists a y such that $x=y+z$.
Negation: Let $x,y,z \in \mathbb{Z}$. There exists an x such that for all y, $x \ne y+z$.

7) There exists unique x such that P(x) us true.
Negation: If x satisfies P(x) then there is a y distinct from x which does too.

8) All mathematics students are hardworking.
Negation: There exists a mathematics student who is lazy.

9) Only some of the students of the class are here today.
Negation: Every student is not in class today.

10) The number $\sqrt{x}$ is rational if $x$ is an integer.
Negation: There exists an integer $x$ such that $\sqrt{x}$ is irrational.I know that this is quite a bit, but I want to make sure that I have negation down.

 

[Ray Vickson]

Homework Statement

Homework Equations

The Attempt at a Solution

1) There exists a grey cat.
Negation: Every cat is not grey.
o
o
o
I know that this is quite a bit, but I want to make sure that I have negation down.

For (1), I would say that the negation of "There exists a grey cat" is "there does not exist a grey cat = there are no grey cats".

 

[Mr Davis 97]

For (1), I would say that the negation of "There exists a grey cat" is "there does not exist a grey cat = there are no grey cats".

I agree. Otherwise does it seem good?

 

[Ray Vickson]

I agree. Otherwise does it seem good?

I did not look at all the others: too many questions!

 

[Mr Davis 97]

I did not look at all the others: too many questions!

Could I just get some help on 6) and 7) then? I think those are the two I am most unsure about.

 

[Dick]

Could I just get some help on 6) and 7) then? I think those are the two I am most unsure about.

For 7) suppose there is no $x$ such that $P(x)$ is true?

 

[Mr Davis 97]

For 7) suppose there is no $x$ such that $P(x)$ is true?

All I did was take the statement of uniqueness and negated it, to get $\forall x(P(x) \to \exists y(P(y) \wedge y \ne x))$, and then reinterpreted it in natural language to say "If x satisfies P(x) then there is a y distinct from x which does too." Is this wrong?

 

[Dick]

All I did was take the statement of uniqueness and negated it, to get $\forall x(P(x) \to \exists y(P(y) \wedge y \ne x))$, and then reinterpreted it in natural language to say "If x satisfies P(x) then there is a y distinct from x which does too." Is this wrong?

It seems to me that 7) is equivalent to "The number of values of $x$ satisfying $P(x)$ is 1." How would you negate that?

 

[Mr Davis 97]

It seems to me that 7) is equivalent to "The number of values of $x$ satisfying $P(x)$ is 1." How would you negate that?

Wouldn't the negation be that there are either 0 or more than 1 value satisfying $P$?

 

[Ray Vickson]

Could I just get some help on 6) and 7) then? I think those are the two I am most unsure about.

For (6): it seems to me that the negation of "For all x there exists a y such that x=y+z" would be "There is x such that there is no y giving x=y+z." (In other words, there is an x such that x-z is not an integer.)

 

[Mr Davis 97]

For (6): it seems to me that the negation of "For all x there exists a y such that x=y+z" would be "There is x such that there is no y giving x=y+z." (In other words, there is an x such that x-z is not an integer.)

Is what I did for 6) necessarily wrong though? Or is it just not the best interpretation?

 

[Dick]

Wouldn't the negation be that there are either 0 or more than 1 value satisfying $P$?

That's how I read it.

 

[Ray Vickson]

Is what I did for 6) necessarily wrong though? Or is it just not the best interpretation?

Actually, you may be right. Or maybe not!
Statement: $\forall x \: \exists y \; \text{such that} \: x = y+z$.
Negation of S: $\exists x\: \text{such that}\: \sim[ \exists y \:\text{s.t.} \: x=y+x]$, where $\sim$ denotes negation. So, what is $\sim[ \exists y \: \text{s.t.} \: x=y+x]$?

$$\sim \exists y \: \text{s.t.} \; x = y+z = \forall y ~[x=y+z] = \forall y \;\; x \neq y+z$$

So, your answer is OK, but the one I gave is actually equivalent to it.