I have got a mother or I have got mother?and I have got a big family: mother, father... and me or I?

Question

Answer

Hi,

"I have got a mother." You need the indefinite article.

Similarly, "I have got a big family: mother, father, ... and me." Of course, if they were the subject of the sentence, it would be (for example), "Mother, father, ... and I live in Moscow."

Answer

"I have got a mother."  There are many mothers, and your mother is one of many: a mother.

Doug has answered your second question well. :)

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