Math: Inclusive/Exclusive "or" and When?

[pivoxa15]

When 'or' is used in any mathematics without extra conditions specified, does it always mean inclusive 'or'? i.e. the first, second or both options.

Any situations when it means exclusive? If so which ones?

 

[sutupidmath]

when it is used "or" (v) it means it is completely enough to satisfy the conditions even if we take just one, however if we take both than it is, how to say even better.

 

[sutupidmath]

Any situations when it means exclusive? If so which ones?

i don't really understand what do you mean here??

 

[matt grime]

Or is always inclusive or. If we want exclusive or then we use Xor.

 

[pivoxa15]

Which areas in maths uses Xor apart from say electrical circuit maths.

 

[Crosson]

Which areas in maths uses Xor apart from say electrical circuit maths.

In combinatorics, we often want to break a set into subset where, for each of some properties $$P_1,P_2,...,P_k$$, exactly one of the subsets has property $$P_i$$ for each $$i$$.

If we want exclusive or then we use Xor.

In my experience most mathematicians will say/write "either A or B but not both" or "exactly 1 of A,B or C is true" , but I have never heard or read xor in math outside of what relates to comp sci; there is an unfortunate trend to look down on explicit use of symbolic logic.

 

[matt grime]

Ah, when I said 'we use Xor' I didn't mean we use the symbol, 'Xor', I meant we explicitly say something equivalent to 'Xor'. Such as, in your example 'precisely one of the subsets P_i'. If we want a symbol for Xor, it is usually the $\oplus$ symbol. The reason you don't see Xor in maths papers is the same reason you *shouldn't* see iff, or (.resp), or many other shorthands. It is just *bad* writing. The paradigm should be that you only use shorthands to aid notation, not to shorten sentences.

 

[Crosson]

It is just *bad* writing. The paradigm should be that you only use shorthands to aid notation, not to shorten sentences.

Because mathematical writing is distinct from doing mathematics, the label "bad writing" for works containing abreviations and logic symbols is a values judgement about which most contemporary mathematicians express similar feelings as matt grime.

Personally, I am weary of over used used phrases in math writing, and I am always pleased to see something other than bourbaki rehash. In fact, the only time I see new phrases tends to be in combinatorics/graph theory and mathematical logic, two fields where the bourbaki group had less impact.

I enjoyed an abbreviation in Doug West's graph theory book, he said that Kuratowski's theorem is an example of TONCAS: the obvious necessary condition is also sufficient.

Sadly, my real fear is that most mathematicians avoid symbolic logic because they did not have the patience to learn it properly. They know what the symbols mean, they know about truth tables, and perhaps they know some of the inference rules, but they never cultivated the ability to read and write math in symbols in real-time.

For example, on of the most misused symbols is $$\Rightarrow$$. Suppose we already have theorem A, and we prove theorem B using only theorem A. Most mathematicians attempt to express this symbollically as
$$A \Rightarrow B$$.

Unfortunately, this does not express anything about the proof of B, for it is similarly true to write $$bannana \Rightarrow B$$, that is, because B is true we can correctly say "anything implies B".

There is a symbol for this situation, logicians say that $$A \vDash B$$
which should be read as "A yields B" or "B is derivable from A". Maybe this will help convince you that dislike of symbolic logic often unfortunately comes from a position of ignorance.

 

[matt grime]

You should *never*, even in a computer science paper, write a sentence where you use XOR as in, 'Thus it follows that foo XOR bar holds'. It's just bad English, bad presentation. That is distinct from saying that the symbol XOR should not be used when appropriate. It is just not appropriate to use it as a sentence constructor, for want of a better phrase. Just as I would never write, 'we take the [itex]\oplus[/tex] of the modules' in a sentence.

It might be a 'judgement about which most contemporary mathematicians express similar feelings as matt grime', but it is based upon the necessity to read things in real time. Why overly complicate things with symbols? There are plenty of symbols already required in reading and writing mathematics, so let's not unduly burden the reader with yet another load of inconsisitently used strings, symbols, and acronyms.

The Bourbakists, incidentally, would probably be closer to your view of things. They like writing "now let C be AB4*", without ever defining what AB anything is.

Abbreviations are different: wlogpptassanwmat, for instance is a goodie for teaching analysis.

However, my experience of teaching logic and truth tables has led me to believe that they are entirely useless for teaching mathematics to undergrads. They entirely fail to understand either how to do the questions, and forget that to disprove for all X Y holds it suffice to find one X where Y doesn't hold.

 

[Crosson]

You should *never*, even in a computer science paper, write a sentence where you use XOR as in, 'Thus it follows that foo XOR bar holds'. It's just bad English, bad presentation.

I agree that it shouldn't be used, because I think it is ugly, but I will point out that math has its own definition of good english, as in "let $$m$$ be an integer" --- so much meaning is left out of this phrase that it requires training beyond standard english to understand the author's intent, this phrase is hardly different than a symbol to the uninitiated.

It might be a 'judgement about which most contemporary mathematicians express similar feelings as matt grime', but it is based upon the necessity to read things in real time. Why overly complicate things with symbols?

Symbols allow economy of expression, and from this, economy of thought. I am able to better understand a theorem that is written on one line then on four. Symbolic logic can make an unweildly theorem "bite size".

If you compare modern math research papers with greek writings, it is quite apparent that symbols have made things simpler, rather then "overcomplicating" them. As matt says, symbols do complicate things when you cannot read and write them in real-time, and this supports what I said:

Sadly, my real fear is that most mathematicians avoid symbolic logic because they did not have the patience to learn it properly...they never cultivated the ability to read and write math in symbols in real-time.