# "one of A and B" or "one of A or B"?

Suppose I have two events A and B, and I want to say that exactly one of them will occur.

Which of the following is grammatically correct:

1. "Exactly one of A or B occurs."
2. "Exactly one of A and B occurs."

Doing a search for this question online has only led to a multitude of probability theory assignments, in which both options appear.

• I would use the or. It is very clearly either A or B. You can make sure by saying Exactly one of either A or B occurs which is belt and braces. Feb 21, 2018 at 6:59
• I don't understand how that's a reasonable Question in basic English, let alone has any place being Featured in ELU. Check it by inverting it. Could you reasonably ask “Might exactly one of A and B occur”? Could you reasonably ask “Might exactly one of A or B occur”? Grammatically either might scrape by but there would still be semantics to face, would there not? Feb 21, 2018 at 22:56
• Either this comment will satisfy you or it won't. The construction 'exactly one of A or B occurs' either came from a math class or another language (perhaps both). If you're in a math class, use XOR, and if not, use 'either ... or.' When you eschew 'either' and use logic gates in text, capitalize all the letters in the gate, OR people will misunderstand (they may misunderstand anyway because I used OR not XOR). Feb 23, 2018 at 17:35

You should use "Exactly one of A or B occurs." For example,

Exactly one of smile or wave occurs.

This implies either a smile or a wave will occur. Conversely

Exactly one of smile and wave occurs.

This implies the person/thing will smile and a wave, doing each once.

I would actually suggest that this may be more understandable.

Either A or B occurs (once).

• I disagree with with 'Exactly one of A or B occurs'. In the original question, it is not stated that A and B are mutually exclusive events. Consider the following example: 'Each interviewee belongs to exactly to of the two groups, coffee drinkers and tea drinkers'. If someone drinks coffee, this person belongs exactly to the first group which does not imply that they don't drink tea. Feb 23, 2018 at 18:33
• I was going off the OP's usage of 'occurs', which implies a single exclusive event. Feb 24, 2018 at 18:25
• I think that's precisely the matter being discussed: 'to occur' does not imply only one event. Consider, 'it either rains or it is warm'. Well, both can occur simultaneously and in the same place. Feb 25, 2018 at 11:31

I have taken a mathematical approach, as the concept of and/or lends itself very easily into binary logic gates and elementary set theory.

In the first case A AND B is true only if both A and B are, whereas A OR B is true if either are. Neither consider the case of exactly one of A/B being true - this is covered by the XOR gate, which is true if A is and B isn't, or vice versa.

In English, the XOR gate is called the 'exclusive-or' gate (as opposed to 'inclusive-or'), which might help explain your predicament.

This might explain the awkwardness of the sentence, and perhaps we should say:

"Exactly one of A xor B occurs."

The second approach notices that A and B might not be independent events, for example:

It rains
The dog barks

Although one may happen without the other, they are not exclusive events.

Conmpare with:

I threw a 1
I threw a 6

which doesn't cover all possibilities, and finally the mutually exclusive events, say:

I threw higher than a 1

So the correctness of the grammar depends on the nature of the events.

• In English, pragmatically or means `xor` whenever a contrast is intended, so that's the normal tendency. The question is whether one is using the English meaning, or the logical meaning (the 'legal meaning' is a matter of arbitrary local opinion). Feb 21, 2018 at 16:26
• Unluckily 'XOR' is defined only as a noun and not as a conjunction. Feb 23, 2018 at 18:36

Legally speaking , it should be event A or event B , whichever is earlier, or occurs first. technically the A and B is Inclusive; the phrase A or B is Selective.

As a reader, I find the word 'exactly' troublesome if it is placed first in the sentence. It leads me to expect to read on about why not "approximately A".

@JonMark Perry is right, XOR, was invented to cater for the case described, but has the disadvantage that it is not a word in normal English usage (yet).

My suggestion: "A or B occurs, but not both."

• Seriously?  You look at “Exactly one of  A …”, and you expect that “exactly” is modifying “A” and not “one”? Feb 25, 2018 at 19:02
• Yes, seriously. "Exactly one" is not normal language. I understand it because I am a mathematician, but normal people wonder what is being talked about. What is inexactly one? Feb 25, 2018 at 22:51
• “Inexactly one” is the number of people who can ride a bicycle at once. OK, I am also a mathematician, but I’m also an English speaker, and I would expect most English speakers to understand “exactly one” just as well as they would understand that “approximately 90,000 people can fit in Wembley Stadium” does not refer to approximate people.  … (Cont’d) Feb 26, 2018 at 1:49
• (Cont’d) … “Exactly N” means “N, no more and no less.” Just as people often interpret the conversational “or” to be the mathematical “XOR”, people often assume that an “at least” is implied with any integer (e.g., “you must have two years of experience to apply for this job”, “you must be three feet tall to ride the roller coaster”, or “you must be 18 years old to vote”) — or maybe “at most” (“you’re allowed to have two pets in an apartment”) — and so sometimes you need to remove that ambiguity. Feb 26, 2018 at 1:49

In my numerous courses of probability and statistics, in order to make allusion to a symmetric difference we always used 'either A or B but not both.'

You could also say 'one of A and B will happen and/but only one.' This version is longer but can be used for a larger set, eg 'one of A, B, or C will happen and only one'.

Place a mental point on the red area, you'll see that it belongs to one circle and one circle only out of the two.

• So, are you saying that the answer to the question is “exactly one of A and B”?  I’m asking because I’m having trouble finding your bottom line. Feb 25, 2018 at 18:40
• ‘(Either) A or B but not both’ is my point as stated in bold. Alternatively ‘(one of) A or B and/but only one’ would also be correct and precise. Feb 26, 2018 at 6:23
• Well, be fair.  You have two things in bold: “either A or B but not both” and “one of A and B … and/but only one.” As I discuss in my answer, “either A or B” is probably the best way to express the thought, but that’s not what the question asks.  If your answer to the question is “(one of) A or B and/but only one”, then you should say so in your answer — but I believe that you can’t treat “one of” as a parenthetical.  As in my example, “One of the team will be promoted to team leader.”, you can’t just omit “One of” and say “the team will be promoted to team leader.” Feb 26, 2018 at 6:39
• Consider 'it will rain or it will be warm' the events are not mutually exclusive. You must say 'it will rain or it will be warm but not both.' Starting the sentence with 'either' is optional in this case. Feb 26, 2018 at 12:41
• No, it’s not true that I must say “it will rain or it will be warm but not both.”  Saying “either it will rain or it will be warm” (or “it will either rain or be warm”) is equivalent.  Ending the sentence with “but not both” is optional in this case. Mar 1, 2018 at 0:59

I think the word "Exactly" brings the ambiguity. I think the word "Either" is more clear to understand for an exclusive or approach.

In which case you would use the 'or' conjunctions: "Either A or B" "Either it stops raining or I'm going to sleep"

Add "...but not both" if you want to be super clear.

Edit: John beat me to it, I'll leave this here for the example :-)

• Your first example is...unclear. What does the rain stopping have to do with you going to sleep? What will you do if it does stop raining? You seem to be using an idiomatic expression we often use in humour (or exasperation). But here the discussion seems to require something more explicit (more obviously "one or the other but not both"). A clearer example for this discussion might be, for example, "Either we eat pizza, or we eat curry for dinner tonight." Feb 23, 2018 at 2:24
• Yes, I agree that your example makes much more sense, but I wanted to emphasise that "either" uses the exclusive OR operator, even with two actions that don't necessarily fit together. The sentence doesn't make much sense (hence the confusion), but the reader knows that I'm doing only one of the actions. Feb 23, 2018 at 14:01

The question is not

Which is correct,

1. Either A or B …

or

1. Either A and B …

?

Of course the answer to that question is “either A or B”, and anybody faced with this situation would be well advised to try to rework their sentence to use “either”.

But the question is

Which is correct,

1. One of A or B …

or

1. One of A and B …

?
[optionally preceded by “exactly”]

This sort of situation can easily be generalized to more than two choices; e.g.,

One of the team will be promoted to team leader.

and, unless the team has only two members, we can’t use “either” in this context.  Well, who’s on the team?  Tom, Dick, and Mary.  So the sentence becomes

One of Tom, Dick, and Mary will be promoted to team leader.

So why would it be any different when there are only two options?

"Exactly one of A or B occurs."... would be the grammatically correct way to express the question clearly to the person who needs to answer the question.

Definition of exactly 1 a : in a manner or measure or to a degree or number that strictly conforms to a fact or condition it's exactly 3 o'clock these two pieces are exactly the same size b : in every respect : altogether, entirely that was exactly the wrong thing to do not exactly what I had in mind 2 : quite so —used to express agreement Meriam Webster

As the question itself states only one event will occur EXACTLY at the same moment; therefore the answer would be one where the outcome A AND B cannot exist at the same moment thus the person answering the question must choose ultimately which alternative meets the criteria in the end. The answer to the question regarding which event happens first is irrelevant to the grammatical structure of how the question is asked. For the question to be expressed clearly to the receiver, the question itself must refer to the answer as one including an alternative, which is the purpose of using the "or" versus the "and" within the question to narrow the answer to EITHER A or B.

• You seem to be placing more emphasis on the word “exactly” than is warranted.  Would you answer differently if the choices were “one of A or B” vs. ”one of A and B” (without the “exactly”)?  Do you have a problem with the phrase “one of them”?  What does the word “them” make you think of?  Answer: it’s a plural pronoun.  How do you construct a plural noun? Answer: “A and  B”. Feb 25, 2018 at 17:35