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The Amora'im disagree about whether Beis Shamai actually allowed the Tzaros of an Ervah to do Yibum. (1)
A Bas Kol from heaven declared that the Halachah follows Beis Hillel.
Even according to the opinion that Beis Shamai did act upon their words, it is not regarded as dividing Klal Yisrael into factions, because Beis Shamai and Beis Hillel had separate Batei Dinim. (2)
In Rebbi Eliezer's place of residence, the people would cut down trees on Shabbos for the purpose of making coals for use in the manufacture of a blade to be used for a Bris Milah on that day. (3)
In the place of Rebbi Yosi ha'Glili, they ate poultry together with milk. (4)
Despite their dispute regarding Tzaros Ervah, the families of Beis Shamai and Beis Hillel intermarried with each other.
Rebbi Avahu maintains that it is permitted to handle an oil lamp on Shabbos, but out of respect for Rebbi Yochanan he did not do so in his presence.
Beis Shamai and Beis Hillel agree that only a child born from a union that is punished with Kares has the status of a Mamzer.
Although Beis Shamai and Beis Hillel disagreed about Tum'ah and Taharah, that did not stop them from handling Taharos together with each other. (5)
Rebbi Akiva maintains that a child born from a union that is punished with a Lav and not punished with Kares has the status of a Mamzer.
Rebbi Shimon says that Beis Shamai and Beis Hillel married into each other's families unless they were notified that a particular match is prohibited to the other.
A BIT MORE
1. The Amora'im who say that Beis Shamai did not act upon their words maintain that this was either because Beis Hillel was the majority or because of the Bas Kol. The Amora'im who say that they did act upon their words believe that this was because Beis Shamai were sharper than Beis Hillel, and -- even though there was a Bas Kol -- Rebbi Yehoshua says that we pay no heed to a Bas Kol.
2. Rava: The prohibition of dividing into factions applies only if there is a split in the same Beis Din. A split between one Beis Din and another is permitted. Abaye: It applies even when there is a split between one Beis Din and another if they are in the same city.
3. Rebbi Eliezer: It is permitted to do a Melachah on Shabbos for the preparation of the Bris Milah. Rebbi Akiva: Any Melachah that could have been done before Shabbos may not be done on Shabbos. This is not regarded as a split in Klal Yisrael, even though Shabbos is very stringent, since the split was in separate places.
4. Rebbi Yosi ha'Glili maintains that the prohibition against eating meat and milk together applies only to animal meat.
5. Out of respect for each other, they notified each other if something was Tamei according to the opinion of the other.
MAKING A KNIFE FOR THE BRIS MILAH ON SHABBOS
The Beraisa says that where Rebbi Eliezer lived, the people would cut down trees for the purpose making coals for use in the manufacture of a blade to be used for a Bris Milah on that day. The Beraisa implies that this was a common occurrence in that place. Why was such an occurrence so common? A Bris Milah on Shabbos is an event that they would know about beforehand and be able to make preparations before Shabbos! Tosfos explains that they intentionally arranged that no knife should be available because they wanted the merit of the precious Mitzvah of doing the preparatory Melachos for the Bris Milah.
HAIRCUTS DURING SEFIRAS HA'OMER
Some places have the custom to take haircuts until Rosh Chodesh Iyar. In those places, the people should not take haircuts after Lag ba'Omer, but on Lag ba'Omer itself it is permitted. In places where the Minhag is to permit haircuts after Lag ba'Omer, they may not take haircuts after Pesach at all until Lag ba'Omer. However, there should not be some people taking on one Minhag and other people taking on the other Minhag in the same place because of the prohibition of "Lo Sisgodedu." Certainly, no one is allowed to take haircuts at both of these times, both after Pesach until Rosh Chodesh Iyar and also between Lag ba'Omer until Shavuos, since these two leniencies are contradictory. (Shulchan Aruch OC 493:3, Mishnah Berurah)
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