What does "opposite" mean?

In a mathematical sense, the opposite of "X" is "not X" and this works in all cases.

But in language text books, or in common usage, there is a lot of ambiguity in "opposite".
Eg.
Father ~ Mother
Son ~ Daughter
Here, both are with reference to the speaker, meaning "my father" ~ "my mother".

Husband ~ Wife
Here , the reference point itself changes, meaning I can not have both husband and wife.

More-over, why should only the sex change for opposite ? Can Son be the opposite of Father ? Considering reversal of both characteristics, we should have Father as the opposite of Daughter, in line with the Dictionary meaning of opposite "Being directly across from each other; facing" or "Characterized by opposite extremes; completely opposed" [from wordweb].

Friend ~ Enemy, Acquaintance, Stranger ?
Uncle ~ Aunt, Nephew, Niece ?

It seems more logical (unambiguous) to use the mathematical sense : opposite of "Father" is "not Father".

In language learning books, what does "opposite" really mean ? Is it some arbitrary colloquialisms ? Is there some unambiguous meaning ?

This question on Linguistics Stack Exchange addresses this, asking if mother and father are true antonyms. I'll quote the accepted answer, written by robert, which basically boils down to "Technically no, but sometimes yes."

Mother is not the antonym of father. They are co-hyponyms because they are both a kind of parent - and parent is the hypernym of mother and father.

Antonymy is the relation that holds between parent and child. So by extension the antonym of mother could be said to be child.

EDIT: After reading hippietrail's answer, I somewhat changed my opinion. If one considers two words to be antonyms already if just one of their semantic features is replaced with the opposite then mother (female parent) and father (male parent) can be said to be antonyms. However, I feel that it might make sense to reserve antonymy for complete negation or oppositeness, and describe the relation between mother and father as hyponymy.

One thing to keep in mind is that language textbooks are often geared toward students who are at a very basic level of that language. For someone who is just beginning to learning English, framing words like mother and father or son and daughter as opposites is a simplification that helps students learn the proper use of those terms. In a simplified way, it makes sense: a mother is a parent who is not a father, and a father is a parent who is not a mother; a son is a child who is not a daughter, and a daughter is a child who is not a son.

This simplification serves the same purpose as the simplification often taught to children just beginning to learn division that even numbers can be divided by two and odd numbers can't. Later on, when they're more advanced, the students will learn that odd numbers can be divided by two; it's just that doing so doesn't result in a whole number. At a more advanced stage, students will learn that words like father and mother aren't true opposites like hot and cold are, but at the moment, that terminology suffices.