"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.

Solution 1:

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).

Solution 2:

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.

Solution 3:

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.

Solution 4:

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."