The prohibition against intermarriage is clarified with the words, “You shall not give your daughter to his son, and his daughter, shall not be taken for your son.”
The very next verse continues, “When your son shall be removed from Me, and he worships false gods, my anger shall be aroused against him.”
The Talmud in Masechet Kiddushin connects the two verses, based on a tradition handed down from G-d to Moshe. When the Torah is specific in stating that “your son turns away,” it is teaching that the offspring of a Jewish father and non-Jewish mother, is “her son,” and not “your son.” Only the son of a Jewish mother, is “your son.” This is the source for matrilineal descent. If we believe in the authenticity, of the Oral Law, then this point is accepted as indisputable. The definition of a Jew, is one born of a Jewish mother.
In recent years, when the number of converts has increased dramatically, there is a concept of זרע יהודי, meaning, “Jewish sees.” The Beit Din, Jewish court, may show some leniencies in allowing one with Jewish relatives to go through the conversion process a little more quickly. This does not mean that such an individual gets a pass, and does not have to accept complete religious observance. However, there is a recurring scenario in today’s world, where all a person may have ever known was Judaism. At a later age, he is shocked to learn that he is not Halchacally Jewish. The Jewish court might allow a quicker conversion, as he was never exposed to any other religion. In any case, we must remember this expression, “her son and not your son.”