"You are Jewish if your mother was Jewish."
1) This is nowhere to be found in the Bible. All the Bible's genealogies are lists of fathers, not mothers.
2) If this were so, that one traces his "Jewishness" through his mother, then none of the Israelites since the Patriarchs were "Jewish" since Judah and his brothers all married Canaanite women with the exception of Joseph who married an Egyptian, (daughter of a priest of Egyptian gods, no less). So, if "Jewishness" comes from one's mother, none of the Israelites were Jewish, they were all descended from Gentile, non-Jewish mothers, (Canaanites and Egyptian).
3) Esau (Edom) is not regarded as "Jewish" even though he had the same mother (and father) as Jacob (Israel).
4) Evidently, "Jewishness" does not automatically come through one's mother.
"You are Jewish is your father was Jewish."
1) Ismael had the same father as Isaac but is not regarded as "Jewish."
2) Evidently, "Jewishness" does not automatically come through one's father.
"You are Jewish is either your mother or your father was Jewish."
1) If all it takes to be reckoned as "Jewish" is for one of your parents to be Jewish then virtually everyone on the planet may be reckoned as Jewish. How? Let's do the math:
2) you are Jewish if 1 of your 2 parents are Jewish
3) 1 of your 2 parents are Jewish if 1 of their 2 parents are Jewish
4) so you are Jewish if 1 of your 4 grandparents, or 1 of your 8 great-grandparents, 16 great-great-grandparents, etc.
5) so all it takes for anyone on the planet to be Jewish is to have 1 ancestor that was Jewish.
6) everyone is descended from up to 277 unique great-great-great...grandparents who lived about the time the temple was destroyed with the genealogical records of the Jews, c. A.D. 70. (About 25 years between an ancestor's birth and begetting the next generation, so 4 generations of ancestors per 100 years, or about 77 complete generations between A.D. 70 and 2010.) Only strict inbreeding within the ancestral tree can reduce this 277 appreciably. But the tendency to inter-breed with other races could not be stopped among Jews in the Bible, much less among the diaspora of Jewish people and proselytes dispersed throughout the world since earliest times. The Pharisees were known for traveling over land and sea to make a single convert. The Talmud has many instructions regarding proselytes and proselytizing. Most probably every single person on the planet has at least one ancestor in his family tree who was Jewish: its just a matter of going back far enough.
7) Evidently, "Jewishness" does not automatically come through one's ancestry.