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 Not necessarily.  Although that depends on how each child was educated at home and what examples they had.

As a couple, men and women must contribute equally to household income.  If the man feels the need that he should be the one who must provide the majority of income for the home, it is because of the way in which he was educated in his home as a child.  Many people believe and even think that the man is the head of the household and has to take responsibility for the whole household, besides making all the decisions in the home, and that thought is not the most appropriate, since as a couple, the husband and the wife are there to support each other.

No, not at all. I believe it is important for both parents to agree on how to provide the income for a family. If that is done by the woman, while the man takes care of the children, and the household chores, supporting the woman in her job, that is equally fine as it being the other way around, or if it is a 50/50 division.