1) WHEN BEIS SHAMAI AGREES THAT A NIDAH IS "TAMEI" RETROACTIVELY
QUESTION: Beis Shamai in the Mishnah (end of 71b) states that if a woman saw a flow of Zov on the eleventh (last) day of her days of Zivah, immersed in the Mikvah the following night, and had relations with her husband, she has the Tum'ah of Mishkav u'Moshav and must bring a Korban. Beis Hillel maintains that she is exempt from a Korban. RASHI writes that Beis Hillel agrees that she is Tamei mid'Rabanan.
The Gemara cites a Beraisa that relates that Beis Shamai argued that Beis Hillel should be consistent: if they agree that she is Tamei, then they should also agree that she must bring a Korban. Beis Hillel replied that Beis Shamai's position is self-contradictory, because Beis Shamai themselves say that if she went to the Mikvah on the following day (the twelfth day), had relations, and afterwards saw blood, she has the Tum'ah of Mishkav u'Moshav but is exempt from a Korban.
The Gemara later quotes Rav Huna who says that according to Beis Shamai, if a woman had a flow of Zov during her eleven days of Zivah (but not on the eleventh (last) day) and went to the Mikvah on the following day, she is Metamei Mishkav u'Moshav even if she did not see at all on the second day (see Rashi DH Amar Rav Huna). The reason for this is because if she would see after going to the Mikvah, she would be Tamei mid'Oraisa, because the flow of Zivah on the second day combines with the Zivah on the first day. Therefore, even when she does not see, there is still a Gezeirah d'Rabanan that she is Tamei.
Rav Yosef questions what new law Rav Huna is teaching. The Mishnah explicitly states that if she immersed on the twelfth day and had relations that night, and afterward she saw Zivah, according to Beis Shamai she is Metamei Mishkav u'Moshav!
Rav Kahana answers that Rav Huna is not teaching the same law as the Mishnah. The Mishnah discusses a case in which she saw blood. Rav Huna, in contrast, discusses a case in which she did not see blood. Rav Yosef retorts that even though in the Mishnah's case she saw blood, nevertheless she saw Dam Nidah (because it was after the eleven days of Zivah). RASHI (DH Re'iyas) writes that since she saw Dam Nidah this cannot combine with the Zivah of the first eleven days. She is Tamei as a Nidah from now on (that is, she is Tamei me'Es l'Es for the first twenty-four hour period before she saw).
How can Rashi write that the Nidah is Tamei me'Es l'Es? The Gemara here is explaining Shamai's opinion (2a) that all women who see Dam Nidah are "Dayan Sha'atan" -- they are Tamei from the time that the see blood and onwards, and not retroactively!
(a) The CHIDUSHEI CHASAM SOFER answers this question based on the words of the Gemara earlier (3b) where Rava says that the reason Shamai is not stringent is because of the Mitzvah of Piryah v'Rivyah. If we would rule that the woman is Tamei retroactively, there would be more abstinence, causing a decrease in the number of children born. However, in the case of the Mishnah, this reason does not apply, because the Mishnah states that Beis Hillel agrees that the woman who goes to the Mikvah on day 12 should not be with her husband on the same day (and a man who is with his wife at that time is called a "Gargaran," a person "greedy" for sin; even though she is not Tamei if she does see blood afterwards on day 12, he is acting wrongly since he may become accustomed to conducting himself in a similar way during the eleven days as well). Since everyone agrees that the woman should not be with her husband on day 12, it follows that Shamai, who generally is concerned about Piryah v'Rivyah, also maintains that she is Tamei retroactively me'Es l'Es.
(b) The author of the footnotes to the CHIDUSHEI HA'RASHBA (2a, footnote 1) suggests another answer. There is a difference between the opinion of Shamai, mentioned in the first Mishnah of Nidah (2a), and Beis Shamai, his students, who maintain, according to Rashi here, that the woman is Tamei retroactively me'Es l'Es. (The CHASAM SOFER also suggests this possibility, but he rejects it as forced.)
Support for this suggestion can be found in an emendation of the RASHASH to Rashi earlier (2b, DH u'Mai Shena), who mentions that the Gemara "asks a question on Beis Shamai." The Rashash amends the words of Rashi to read that the Gemara "asks a question on Shamai." Perhaps the Rashash makes this change because it indeed is only Shamai who maintains, in the beginning of Nidah, that the woman is not Tamei retroactively, while we see from Rashi, in the end of Nidah, that Beis Shamai, the students of Shamai, disagree and maintain that she is Tamei retroactively. (D. BLOOM)
2) THE ELEVENTH DAY OF THE DAYS OF "ZIVAH"
QUESTION: Rebbi Elazar ben Azaryah rules that the period of eleven days of Zivah between each period of Nidah is a Halachah l'Moshe mi'Sinai. The Gemara asks, "What is this Halachah?" Rebbi Yochanan answers that it is the "Halachah of the eleventh day," referring to the Halachah that the eleventh day does not require Shimur. Reish Lakish answers that it is the "Halachos of the eleventh day," referring to two Halachos -- the eleventh day does not require Shimur, and the eleventh day does not serve as Shimur for the tenth day (that is, the tenth day does not require Shimur).
How do Rebbi Yochanan and Reish Lakish infer from the words of Rebbi Elazar ben Azaryah that a specific Halachah concerning the eleventh day is a Halachah l'Moshe mi'Sinai? Rebbi Elazar ben Azaryah simply states that the period of eleven days of Zivah is a Halachah l'Moshe mi'Sinai -- that is, the entire concept of the eleven-day period is a Halachah l'Moshe mi'Sinai, and not just the eleventh day!
If Rebbi Elazar ben Azaryah's statement refers specifically to laws of the eleventh day, what is the Gemara's question that these laws are derived from verses and are not Halachos l'Moshe mi'Sinai? The law of the eleventh day does not appear in any verse! (TOSFOS DH Rebbi Yochanan Amar)
(a) TOSFOS explains that Rebbi Elazar ben Azaryah's Halachah refers to the entire period of eleven days of Zivah. Rebbi Yochanan and Reish Lakish are arguing about another Halachah, one related to the teaching of Rebbi Elazar ben Azaryah. They are not explaining his statement.
(b) RASHI explains that Rebbi Elazar ben Azaryah refers specifically to the eleventh day. Rashi seems to understand that Rebbi Elazar ben Azaryah is teaching two Halachos l'Moshe mi'Sinai. This is inferred from his words, "The [period of] eleven days between each [period of] Nidah is a Halachah l'Moshe mi'Sinai." Since he did not say simply, "The days between each [period] of Nidah," it is clear that he is emphasizing that there are two parts to the Halachah l'Moshe mi'Sinai: first, that there is a period of eleven days, and, second, that the eleventh day is unique among them (the subject of the argument between Rebbi Yochanan and Reish Lakish). (M. KORNFELD)