Statement -> Conditional, Converse, Inverse, Contrapositive

Gigazombie Cybermage

How do you turn a normal statement into a conditional?

"All stores are closed."

I thought the conditional would be: If they are all stores, then they are closed.

Here's what I got for my other answers.

Converse: If they are closed, then they are all stores.
Inverse: If they are not all stores, then they are not closed.
Contrapositive: If they are not closed, then they are not all stores.

I also have to show which of them is logically equivalent to the original statement. Not sure if he wants a truth table showing it or not, doesn't really give me room on the sheet to do it though.

kime

That is a weird question. Like, typically for conditionals you need two statements, essentially. So I get that's why you've sort of broken it up to "all stores" and "are closed," but that just seems like a strange phrasing. (Are you sure there's not a second statement in the question somewhere? )

But if you treat those as your two statements, then yeah, your other answers look right. The only way to really show which of them are equivalent is with a truth table, I think. So if you just need to give the answer, you can do the table on another page. If you have to show it, then write small :P

To help, it can all be in one truth table, you know that yeah? You don't need a separate table for each thing, just put them in different columns.

Lachrymite

If X is a store, then it is closed?

kime

Lachrymite wrote: »

If X is a store, then it is closed?

Oh yeah, that sounds a lot better :P

TychoCelchuuu

First we turn it into an if... then (->) statement:

If it is a store, then it is closed. Or store -> closed.

The converse is just turning it around:

Closed -> store (if it is closed, then it is a store).

For the inverse we put "not" in front of both parts of the original:

Not store -> not closed (if it's not a store then it's not closed).

For the contrapositive, we reverse the statement and also add nots:

Not closed -> not store (if it's not closed then it's not a store).

I don't know why people bother teaching this stuff because it's pretty worthless but there you go. The only useful part is turning the original statement into a conditional but it looks like they didn't really bother teaching you how to do it. I think the best way to think about it is, "what information does the sentence give me?" Well, if all stores are closed, then we know for every X, if X is a store, then X is closed. We also know the contrapositive (for every X, if X is not closed, then X must not be a store, because all stores are closed) but presumably they wouldn't like it if you just translated the sentence directly into the contrapositive even though they are logically equivalent.

Gigazombie Cybermage

That's what I thought, thanks guys . Yeah, I'm sure there isn't another statement, that's why the phrasing I put it in is kinda weird. That "all" in there just makes it weirder to do it correctly.

Gigazombie Cybermage

Oh, one more thing. The negation for "All... are not" is "Some... are not" right?

TychoCelchuuu

The negation of "for all X, X is Y" is "it is not the case that for all X, X is Y" which is logically equivalent to "there is some X such that X is not Y." If you have to actually learn this stuff rather than just regurgitate it once in the homework and never again I think you will be well served by actually trying to understand the logic behind the various transformations rather than memorizing which things transforms into which.