Translating set notation to english

[spaghetti3451]

I've been trying to think of the grammatically correct way to translate A\cupB and A\capB.

So, let's say A is the set of all animals and B is the set of all boats.

Then, A\cupB is the set of all entities which are either animals or boats (or both).

And A\capB is the set of entities which are both animals and boats.

I'm wondering if there's any ambiguity in my translation now that I've used very precise wording. What do you think?

 

[economicsnerd]

That looks perfectly correct/precise to me.

 

[micromass]

I'm not an expert on english language by far, but doesn't "either" specifically mean one or the other and not both? This is what I always thought. I would appreciate some native english speakers to shed a light on this.

 

[jbunniii]

I'm not an expert on english language by far, but doesn't "either" specifically mean one or the other and not both? This is what I always thought. I would appreciate some native english speakers to shed a light on this.

Often but not always.

"You must choose either me or him" = either one or the other, not both
"You can run this software on either Mac or PC" = it works on both Mac and PC

One can argue that the second example should be worded more carefully, but it's a common usage.

 

[micromass]

Often but not always.

"You must choose either me or him" = either one or the other, not both
"You can run this software on either Mac or PC" = it works on both Mac and PC

One can argue that the second example should be worded more carefully, but it's a common usage.

Great! Thanks a lot!

 

[Mark44]

I'm going to disagree - either A or B implies one or the other, but not both. jbunnniii's example might be common usage, but lots of common usage is not grammatically kosher, such as "who do you want to see". The intent of the Mac/PC example is "You can run this software on a Mac or a PC."

 

[jbunniii]

The English language just isn't as precise as we would like sometimes. Rules have exceptions, and the rules (e.g. dictionary definitions) evolve over time to reflect actual usage.

To make matters even more confusing, sometimes "either" is used to mean "both":

"Either way is correct" = both ways are correct
"There are trees on either side of the road" = there are trees on both sides of the road

 

[Mark44]

I agree that English isn't precise, but still maintain that the most common meaning of sentences containing "either <this> or <that>" is the exclusive or, as in, "On our vacation this year, we will be going either to the beach or the mountains." The clear intent is that we will be going to one of those places, but not both.

 

[jbunniii]

I agree that it's the usual meaning. But as in mathematics, finding counterexamples can be the fun part.

 

[Fredrik]

I agree with jbunii's usage example, but I want to add that in mathematics the phrase "either p or q" means specifically that exactly one of p and q is true. A similar thing is that outside of mathematics, a phrase like "there are two letters in the mailbox" would mean that there are exactly two letters in the mailbox, but in mathematics, "there are two integers with property p" doesn't exclude that there are three.

 

[disregardthat]

I've never used either/or as exclusive or, if I say "either p or q" I would allow for both p and q to be true.

Take a look at this: http://en.wikipedia.org/wiki/Logical_disjunction

The "or" operator differs from the exclusive or in that the latter returns false when both of its inputs are true, while "or" returns true. In ordinary language, outside of contexts such as formal logic, mathematics and programming, "or" sometimes has the meaning of "exclusive or". For example, "Please ring me or send an email" likely means "do one or the other, but not both". On the other hand, "Her grades are so good that she's either very bright or studies hard" allows for the possibility that the person is both bright and works hard. In other words, in ordinary language "or" can mean inclusive or exclusive or. Usually the intended meaning is clear from the context.

In this example they do not equate either/or with exclusive or.

 

[Mark44]

The fattest dictionary I own is Webster's Encyclopedic Dictionary of the English Language. The first entry for either is, "1. one or the other of two."

The entry for either-or is "allowing no equivocation; being limited in choice to two options: It's an either-or situation- you pay the bill or you lose the company's services."

The example above (in italics) and the "allowing no equivocation" definition suggests to me that they're talking about mutually exclusive options.

Another dictionary I have, The American Heritage Dictionary of the English Language, New College Edition, starts off with this definition: "One or the other: Choose either. Either will serve the purpose."

Later in the definition, they write, "...Similarly, the conjunction either, when used in combination with or, is most apropriate to statements involving two alternative elements."

An example they give - "Either he obeys or he leaves." is one where the alternatives are mutually exclusive.

From the Oxford Dictionaries (http://www.oxforddictionaries.com/definition/english/either - I don't know if this is what the OED is now being called) entry for either, "either I accompany you to your room or I wait here" and "available in either black or white". Both examples have options that are mutually exclusive.

In the example that disregardthat quoted from wikipedia -- "Her grades are so good that she's either very bright or studies hard" -- there seems to be an implied and unstated "or both" at the end that would permit the possibility of both conditions being true simultaneously.

 

[disregardthat]

Examples such as "either I accompany you to your room or I wait here" where the options are exclusive by themselves doesn't argue for that it's the word "either" that makes it so. You can't logically do both anyway. To exemplify, one needs a statement with options not mutually exclusive, in which "either" makes it so, since "either" can of course be used in situations where the options by themselves are mutually exclusive. It's also true that in certain contexts, mutual exclusivity is implied, using the word "either" or not.

But I don't see these dictionaries implying that either/or logically means exclusive or. To me, it rather seems to highlight the fact that the two options are the only ones available (which is particularly true in situations such as "p or (not p)"), not that they are exclusive options.

 

[darkchild]

I'm wondering if there's any ambiguity in my translation now that I've used very precise wording. What do you think?

It's clear. You could possibly make it a bit more precise by specifying the universe of discourse, particularly since the sets A and B are made up of such disparate entities.

 

[disregardthat]

There's a discussion about the subject here:

http://english.stackexchange.com/questions/13889/does-either-a-or-b-preclude-both-a-and-b

They also make a point there that in ordinary language, emphasis on the word "either" usually means the intention is an exclusive or.

 

[Mark44]

Examples such as "either I accompany you to your room or I wait here" where the options are exclusive by themselves doesn't argue for that it's the word "either" that makes it so. You can't logically do both anyway.

That's true, but I believe the use of "either" with "or" emphasizes that the choices are mutually exclusive.

To exemplify, one needs a statement with options not mutually exclusive, in which "either" makes it so, since "either" can of course be used in situations where the options by themselves are mutually exclusive. It's also true that in certain contexts, mutual exclusivity is implied, using the word "either" or not.

But I don't see these dictionaries implying that either/or logically means exclusive or.

The references I quoted strongly suggest (to me at least) that the sense is exclusive or.
Webster's Encyclopedic Dictionary of the English Language: "1. one or the other of two."
The American Heritage Dictionary of the English Language, New College Edition: "One or the other: Choose either. Either will serve the purpose."

To me, it rather seems to highlight the fact that the two options are the only ones available (which is particularly true in situations such as "p or (not p)"), not that they are exclusive options.

Spoken language is difficult to pin down, because it is much less well-defined than, say mathematics or symbolic logic expressions, where we distinguish between inclusive or and exclusive or. What I am saying is that in the majority of sentence constructions using "either ... or", the intent is to indicate that the choice is between one of two mutually exclusive options.