Statement -> Conditional, Converse, Inverse, Contrapositive

Gigazombie CybermageGigazombie Cybermage Registered User, __BANNED USERS regular
edited October 2012 in Help / Advice Forum
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.

Gigazombie Cybermage on

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  • kimekime Queen of Blades Registered User regular
    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.

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  • LachrymiteLachrymite Registered User regular
    If X is a store, then it is closed?

  • kimekime Queen of Blades Registered User regular
    Lachrymite wrote: »
    If X is a store, then it is closed?

    Oh yeah, that sounds a lot better :P

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  • TychoCelchuuuTychoCelchuuu PIGEON IndiaRegistered User regular
    edited October 2012
    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.

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  • Gigazombie CybermageGigazombie Cybermage Registered User, __BANNED USERS regular
    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 CybermageGigazombie Cybermage Registered User, __BANNED USERS regular
    Oh, one more thing. The negation for "All... are not" is "Some... are not" right?

  • TychoCelchuuuTychoCelchuuu PIGEON IndiaRegistered User regular
    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.

    kimeEssee
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