Is there any type of objects or ideas for which asking about their equality makes sense, but asking about non-equality doesn't? (or vice versa)
Intuitively, "not equal" is a negation of "equal", so my guess is that this is actually asking about the same thing, just from different perspectives. And personally I cannot imagine anything for which the question about its being equal to something else of the same kind be possible, but asking about the opposite wasn't. But what if my imagination is limited or ignorant of something? Mathematics is based on proofs, not one's ability to imagine something. So it would be better to have a proof.
To make things straight:
My concern is not about whether they're mutually exclusive or not – this is pretty much obvious, since things cannot be equal and non-equal at the same time.
I'm rather asking about the very possibility of asking questions about the latter and the former. Are there any object $a$ and $b$ for which asking if $a = b$ is true does make sense, but asking if $a \neq b$ is true doesn't? (Similarly to the case where it does make sense to say that they are non-equal, but does not make sense to ask whether one is greater than or less than the other, as we have for non-ordered sets).
Or, in other words: $=$ and $\neq$ are binary operators which have some logic value (truth or false) when applied to some objects. Are there any objects for which the operator $=$ is defined, but the operator $\neq$ doesn't?
My question follows from the observation that in some programming languages one can define custom operators $=$ and $\neq$ for user-defined data types, but these operators are defined separately: one can define one but not the other, or both. So I wonder if there are any cases where defining one but not the other actually does make any sense.