(Reish Lakish): If one ate Pigul and Nosar (and Tamei - some texts omit this) that were mixed together, he is exempt. Surely, one of them was the majority, and the other is Batel.

Inference: If an Isur gives taste to a food of a larger volume than itself, it does not forbid it (to Mechayev lashes) mid'Oraisa (Reish Lakish's law applies even when the minority gives taste),

80a (Mishnah - R. Eliezer): If a small amount of water fell into a flask with Mei Chatas (water on which ashes of the Parah Adumah were put), for every sprinkling done from this flask (to be Metaher), two Haza'os must be done (this will be explained);

Chachamim disqualify (the water may not be used).

Chachamim hold that (even if) Yeish Bilah (we assume that one liquid totally disperses amidst another), there is a Shi'ur (minimum required quantity) for Haza'ah, and we cannot combine Mei Chatas from different Haza'os to comprise the Shi'ur;

Bechoros 22a (Reish Lakish): If one buys brine from an Am ha'Aretz (who is not trustworthy about Taharah), he may do Hashakah (connect it to a Mikvah) to be Metaher it, in any case;

If the majority is water, Hashakah is Metaher water. If the majority is brine, brine is not Mekabel Tum'ah, and the water is Batul in the majority.

(R. Yirmeyah): This is Metaher the brine to use for a dip (without adding water). If he will (add water, e.g.) cook it in a pot, and altogether there will be more water than brine, this arouses the Tum'ah.

Menachos 23a (Rav Chisda): If a Neveilah became mixed with a majority of Shechutos (slaughtered animals), it is Batel, for a Shechutah cannot become (an Av ha'Tum'ah, like) a Neveilah. (This is like Min b'Eino Mino.) If a Shechutah became mixed with a Neveilah, it is not Batel, for a Neveilah can become (Tahor) like a Shechutah, i.e. when it rots (so this is like Min b'Mino).

(R. Chanina): If the Batel (minority) can become like the Mevatel (majority), it is not Batel; if the Batel cannot become like the Mevatel, it is Batel

These teachings are not like Chachamim, who say that Min b'Mino is (always) Batel. They are not like R. Yehudah, who says that Bitul depends on the appearance (22a; Dam Par cannot Mevatel Dam Sa'ir, even though neither can become like the other). Min b'Mino is never Batel, whether or not one can become like the other! They are like R. Chiya;

(Beraisa - R. Chiya): If Shechutos and Neveilos became mixed, one of these (if it is the majority) is Mevatel the other.

Question: R. Chiya is unlike Chachamim, who say that Min b'Mino is (always) Batel. He is unlike R. Yehudah, who says that Min b'Mino is not (i.e. never) Batel!

Answer: He is like R. Yehudah. Really, R. Yehudah says that Min b'Mino is not Batel only when one can become like the other, but if one cannot become like the other, there is Bitul;

Rav Chisda says that it depends upon the Mevatel (if it cannot become like the Batel, there is Bitul). R. Chanina says that it depends upon the Batel (if it cannot become like the Mevatel, there is Bitul).

(Mishnah): If two Menachos became mixed together before Kemitzah, if one can take a Kometz from each of them by itself they are Kesherim. If not, they are Pesulim.

Observation: After taking one Kometz, the rest of that Minchah is Shirayim. (Surely, there are pieces of the Tevel (the Minchah from which Kometz was not taken) amidst a majority of Shirayim. Since both are Kesherim, this shows that the pieces are not Batel. If they were Batel, the Tevel would be considered Chaser before Kemitzah!) This is like R. Yehudah.

Question: Rav Chisda holds that since the Mevatel cannot become like the Batel, there should be Bitul. The second Minchah should be Pasul. He must say that our Mishnah is unlike R. Chiya!

Sukah 9b (Mishnah): If one draped a vine over a Sukah, it is Pasul. If there was more Sechach than vines or if he cut them, it is Kosher.

9b - Question: If he did not do Chavatah, why it is Kosher? Pasul Sechach joins with Kosher Sechach!

Answer: We must say that he did Chavatah.

Nidah 71a (Mishnah - R. Yehudah): If a woman was about to give birth, and a Revi'is came out, it is Metamei due to Kesem (surely, at least a drop left the Makor before death).

R. Yosi says, therefore, it is not Metamei b'Ohel (for there is not a full Revi'is).

Rosh (Sukah 1:14): Rashi says that 'Chavtah' is lowering the vine and mixing it with the Kosher Sechach. The vine is Batel in a majority of Kosher Sechach.

Rambam (Hilchos Sukah 5:16): If one put Kosher and Pasul Sechach next to each other, if there is more Kosher Sechach it is Kosher. If there is exactly the same of both it is Pasul, because Pasul Sechach is like Parutz.

(Magid Mishnah): It is as if the Pasul Sechach is not there, therefore we need a majority of Kosher Sechach.

Tosfos (78a DH ha'Pigul): A minority Batel in a majority does not join with it to complete the Shi'ur (to be liable). Heter does not join Isur, so all the more so Isur does not join Isur!

Tosfos (79b DH Tanan): It is a stringency mid'Rabanan to disqualify Mei Chatas in which a drop of water fell. Mid'Oraisa, it is Batel.

Or Some'ach (Hilchos Tzitzis 1:11): If a string that was not spun l'Shem Tzitzis were mixed with strings spun Lishmah, Acharonim were unsure if it is Batel and one may use them for Tzitzis. Even though we know that one of them was not spun Lishmah. The general rule of Bitul is that it is as if it is not there. It does not join. There is an awesome proof from the end of Nidah. (There is also a proof from Rashi Sukah 47b DH sheha'Yayin. If a snake drank from Nesachim and put in venom, the Kohen will not pour the full Shi'ur of wine. We do not say that through Bitul, it joins to comprise the Shi'ur!) Here also, it is as if the string not Lishmah is not here. Bitul applies only when the Heter does not require an action. E.g. if Neveilah is Batel in pieces of Heter, even though Shechitah requires intent (at least to cut), the reason it is Neveilah goes away, and it is not considered Neveilah nor slaughtered. It is like Stam food. Neveilah can become Shechutah, i.e. if it spoils and a Ger (Toshav) would not eat it (Menachos 23). Even though it does not (truly) become Shechutah, it is not Neveilah. Here, we cannot say that through Bitul the string is Kosher for Tzitzis and it is as if it was spun Lishmah. However, if a little water fell into Chatas, the water may not to be used. Tosfos says that this is a stringency of Parah Adumah, but mid'Oraisa the water is Batel in the majority. Mei Chatas require an action of Kidush (putting the ashes on) l'Shem Mei Chatas. How can we say that through Bitul, it is as if this was done?! If ashes of the Parah Adumah were mixed with plain ashes, we may not be Mekadesh with them. I can explain that this is because the ashes must come from a cow that was burned and done l'Shem Mei Chatas. Bitul cannot cause that the plain ashes were l'Shem Mei Chatas. Tosfos must say that this is a stringency of Mei Chatas. We can say that mid'Oraisa it is Kosher due to the majority. We assume that what is sprinkled came from the majority, and not from the drop of plain water mixed in. However, we cannot say so if Yesh Bilah and sprinklings do not join.

Or Some'ach: Through Bitul, it is as if the minority is not here. This does not help for something that requires a Shi'ur. If there is Bilah, part of the Shi'ur is missing, if sprinklings do not join. Based on what I said, since Kidush was not done, even if Ein Bilah and there is no required Shi'ur, Bitul did not make the water Mei Chatas, so it is like a Safek Kavu'a (an even doubt. We do not say that what we take comes from the majority). The Rambam says that Kavu'a does not apply to something that is not recognizable, because mid'Oraisa even living animals are Batel. However, when there is no Bitul, we do not distinguish whether or not it is recognizable. Therefore, regarding a string not spun Lishmah mixed with strings spun Lishmah, since Bitul does not make it as if it were spun Lishmah, we do not say that what is taken came from the majority, for we take from Kavu'a.

Rashi (Bechoros 22a DH Aval): If brine is mixed with a majority of water, this is Metamei the brine.

Rashba (Menachos 23b DH v'Rebbi): if each of them could Mevatel the other, whether or not it can become like the other, R. Chiya would have said that Min b'Mino is Batel, without specifying Neveilos and slaughtered animals.)

Oneg Yom Tov (4): Bitul removes an Isur, but does not make something considered a Kosher Korban. There is no proof from ashes of Parah Adumah. Tosfos says that this is a mere stringency. There is no proof from R. Eliezer who permits using it, for perhaps he requires sprinkling twice (so surely one sprinkling was with Kosher Mei Chatas).A proof is from Menachos 23. We challenged the opinion that there is no Bitul when the Mevatel can become like the Batel. A Mishnah says that if two Menachos became mixed, if one can take a Kometz from each of them by itself they are Kesherim. The Tevel is not Batel in a majority of Shirayim, even though Shirayim cannot become Tevel! Rashi asked that this is difficult also for the opinion that there is no Bitul when the Batel can become like the Mevatel. Since there is Bitul, the Tevel Minchah has extra flour! Rashi answered that through Bitul, it is as if the Shirayim are not there, so there is no extra flour. Even though the Shirayim are Batel in the Tevel, it is not considered that there is extra flour. The Batel is not considered totally like the Mevatel, only the Isur of the Batel vanishes. Also when we say that Isur becomes Heter, this means only that the Isur goes away. However, Tosfos (22 DH v'Hen) challenged Rashi from the Mishnah of a Kometz mixed with a Minchah of a thicker consistency. The Kometz disqualifies the Mishnah, because the Minchah has extra oil. This shows that the Batel becomes totally like the Mevatel. Also Rashi said so at the beginning of the Sugya, that Neveilah and Shechutah are not Batel in each other, i.e. if later one of them touched Terumah. We do not burn the Terumah. It is considered Safek, for perhaps the Shechutah touched it, i.e. and it was not Batel in the Neveilah. According to the opinion that it depends on the Batel, the Shechutah is Batel in the Neveilah, and we would burn Terumah! However, perhaps Rashi holds that Shechutah does not become like Neveilah even after Bitul. R. Chanina says that we burn Terumah because we attribute that it touched the majority, i.e. a Neveilah. Rav Chisda holds that there is no Bitul, so it is like an Isur Kavu'a. Also Tosfos says that it is considered a Safek Kavu'a only when there is no Bitul. However, in Bechoros Rashi connotes that something Batel becomes like the Mevatel in every way. If it itself does not receive Tum'ah, through Bitul it can receive Tum'ah. Tosfos disagrees, and says that surely the brine is not Mekabel Tum'ah. If so, strings Lo Lishmah mixed with strings Lishmah may not be used for Tzitzis.

Ha'aros on Shirei Berachah (YD 279:3): According to the Rishonim in Sukah who say that through Chavatah the Pasul Sechach becomes Kosher and one may sit under it, especially according to the Bach, Bitul helps also for Lishmah. However, we can say that Sukah does not need Lishmah. It needs only l'Shem shade, and perhaps Chavatah fulfills this! In any case through Bitul, the Pasul Sechach becomes Kosher. However, Sukah 24 shows that not all the Sechach needs to be put Lishmah.

Note - it seems that this should say 15a, in which Beis Hillel say that it suffices to put every other board Lishmah.

Shulchan Aruch (OC 11:1): The strings for Tzitzis must be spun Lishmah.

R. Akiva Eiger (1): If strings not spun l'Shem Tzitzis were mixed with strings spun Lishmah, see 626:3 regarding whether one may use them for Tzitzis. Perhaps this is Davar sheb'Minyan (something important that is sold by number; it is never Batel).

Shulchan Aruch (626:1): If a Sukah has more shade than sun only with (connected branches of) a tree, one must lower the branches and mix them with the Sechach so they will not be recognized, and the Sechach will be the majority and be Mevatel the branches.