Thread: 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?

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.

First, it is important that the original statement



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 , but then one can also choose to apply contrapositive to inner implication . Also, an assumption like for some domain is usually considered almost a part of the quantifier. For example, we usually say that the negation of is and not . Similarly, the usual contrapositive of is and not .

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.

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 is ? This is not saying much. The negation of is . Also, inverse is not the same as negation.

Originally Posted by emakarov

Hmm, do you mean that the negation of is ? This is not saying much. The negation of is . 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.

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