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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.

After some thinking [and the antonyms reference given by user Nicole helped a lot; user Hippietrail seems to support my thinking and has clarified a lot of things], I feel I can chip in with my own perspective on this question.

Antonymy : The semantic relation that holds between two words that can (in a given context) express opposite meanings [From WordWeb, emphasis mine]

Seems to be a good Definition, but this question is not about Antonyms, rather about what "opposite" means.

Opposite : "A word that expresses a meaning opposed to the meaning of another word, in which case the two words are antonyms of each other" & "Altogether different in nature, quality or significance" [From WordWeb, emphasis mine]

So if a word has many characteristics, then there can be many opposites, where atleast one characteristic is "negated".

Eg Father has characteristics (Male,Parent,Relative,Good,Carer,...) so opposites can be Mother (Male is negated Female) or Son (Parent is negated to Child) or Daughter (Male is negated Female & Parent is negated to Child) ...

What would be the opposite of Father in "My class teacher was like a Father who treated us like his children" ? The characteristic is "Good" and negate that to get "Bad", so "My class teacher was like a Tyrant who treated us like slaves". Of course, in general, nobody claims that the opposite of Father is Tyrant.

So I could say that "opposite" is either not really Well Defined, or any usage must state which Specific Context is being considered.
User Hippietrail rightly says that "antonym" is not really a "natural" or "God given" concept that it at first seems.

It may not matter much in daily life, but in language Exams, incorrect thinking on the part of the evaluators may result in incorrect evaluations of some "correct" answers. One way out would be to ask questions like "What is the Feminine opposite of Father ?" so that the context is shown.

Not doesn't mean opposite. Animals that are not birds is the set of all animals minus birds - Not means 'exclusion'.

The opposite of 6 is negative 6 - opposite meaning a measurement of the same magnitude in the reverse direction of the same continuum.

So on the continuum of approval, or morality, good is the opposite of bad.

On the continuum of personal alliances a friend is the opposite of enemy.

But titles such as Uncle are not on any continuum, they have one specific meaning. There is no opposite of Doctor, Dentist, baseball or niece.