Write a negation, contrapositive, converse

Statement: For every rational number z and every irrational number x, there exists a unique irrational number y such that z=x+y.

Write its negation, contrapositive and converse. Simple enough right? NOT SO, according to my teacher.

I think whats messing me up is "For every rational number z and every irrational number x" in the beginning.

Anyone wanna take a crack at it? And can someone give me examples similar to this one?

Re: Write a negation, contrapositive, converse

Think of the statement as an implication, and it is easier. A is the stuff before the comma, B is after it. Then the statement is claiming "if you have case A, then case B is true".

Negation / inverse is the negation of A implies B. Converse is B implies A. Contrapositive is both of these together. You should be able to follow from that, but let me know.

Re: Write a negation, contrapositive, converse

First, it is important that the original statement

$\displaystyle \forall z\in\mathbb{Q}\,\forall x\in\mathbb{R}\setminus\mathbb{Q}\,\exists!y\in \mathbb{R}\setminus\mathbb{Q}\;z=x+y$

uses the uniqueness quantifier. To negate the statement properly, it helps to express the uniqueness quantifier through ordinary quantifiers.

I am not sure about contrapositive and converse, though. Strictly speaking, these concepts apply to implications. The original statement *can* be considered as an universally quantified implication $\displaystyle \forall z\,z\in\mathbb{Q}\to\dots$, but then one can also choose to apply contrapositive to inner implication $\displaystyle \forall z\,z\in\mathbb{Q}\to(\forall x\,x\in\mathbb{R}\setminus\mathbb{Q}\to\dots)$. Also, an assumption like $\displaystyle x\in A$ for some domain $\displaystyle A$ is usually considered almost a part of the quantifier. For example, we usually say that the negation of $\displaystyle \exists n\in\mathbb{N}\,n<0$ is $\displaystyle \forall n\in\mathbb{N}\,\neg n<0$ and not $\displaystyle \forall n\,n\notin\mathbb{N}\lor\neg n<0$. Similarly, the usual contrapositive of $\displaystyle \forall n\in\mathbb{N}\,n\le0\to n=0$ is $\displaystyle \forall n\in\mathbb{N}\,n\ne0\to\neg n\le0$ and not $\displaystyle \forall n\,\neg(n\le0\to n=0)\to n\notin\mathbb{N}$.

Here, when universal quantifiers are stripped from the original statement, we have an existential quantifier, and under it we have a conjunction (or biconditional), not an implication.

Quote:

Originally Posted by

**Annatala** Think of the statement as an implication, and it is easier. A is the stuff before the comma, B is after it. Then the statement is claiming "if you have case A, then case B is true".

Negation / inverse is the negation of A implies B.

Hmm, do you mean that the negation of $\displaystyle A\to B$ is $\displaystyle \neg(A\to B)$? This is not saying much. The negation of $\displaystyle A\to B$ is $\displaystyle A\land\neg B$. Also, inverse is not the same as negation.

Re: Write a negation, contrapositive, converse

Quote:

Originally Posted by

**emakarov** Hmm, do you mean that the negation of $\displaystyle A\to B$ is $\displaystyle \neg(A\to B)$? This is not saying much. The negation of $\displaystyle A\to B$ is $\displaystyle A\land\neg B$. Also,

inverse is not the same as negation.

Ah, my bad. It's been a looooong time since I've used those terms and I'm a little rusty. Inverse is applying negation to *each* side of the implication, then. That makes more sense as the contrapositive needs to be logically equivalent to the original.

Re: Write a negation, contrapositive, converse

It's very easy to get confused. It seems so simple though.