Before us is Mishnah 10 in the fourth chapter of Maseches Nedarim. The Mishnah lists four disputes between Beis Shammai and Beis Hillel, in four separate topics unrelated to one another, and in all of them Beis Shammai are lenient and Beis Hillel are stringent. We will examine each one on its own.
The first dispute: one who forbids his wife to him by a vow regarding marital relations:
A vow is a prohibition that a person places upon a particular thing, as though it were a korban or an object of hekdesh. This prohibition can operate in two directions: a person may forbid his fellow's object to himself ("So-and-so's shoes are forbidden to me"), and he may forbid his own object to his fellow. What he cannot do is forbid his fellow's object to his fellow, since he is not its owner. But his own body is in his hands, and therefore he can declare that he will derive no benefit from another person's property.
Our Mishnah speaks of a husband who forbids marital relations with his wife to himself. He cannot forbid the wife from deriving benefit from him, since a wife has rights in the marriage - food, clothing and marital relations - and this right is hers; just as a person cannot forbid to his fellow that which belongs to his fellow, so too the husband cannot take it away from her. But he may declare that he himself will derive no benefit from his wife, and even though the result is that her rights to marital relations are not fulfilled, the vow takes effect.
The question the Mishnah discusses is: how long can this situation continue before the wife is entitled to a get and to collect her kesubah, because she is not receiving what is due to her by virtue of the marriage? This is the period of time available to the husband to find a solution, such as annulment of the vow. At the end of this period the wife may bring her husband to beis din, and beis din will instruct him: either resolve the matter immediately, or give a get and pay the kesubah.
Beis Shammai - two weeks: They derive this from what we find in the Torah regarding a woman who gives birth to a daughter, who is forbidden to her husband for fourteen days. The husband, as it were, brought this upon himself, and therefore a restriction of marital relations for such a period is within the range of the reasonable. Beyond that, the wife need not be expected to wait, and she can demand a divorce.
Beis Hillel - one week: The birth of a daughter is not a common event, and the framework by which to measure is the ordinary situation - the prohibitions of niddah, which are far more common. By Torah law this is a restriction of seven days (the Sages later extended it), and therefore seven days are all that he is given, and beyond that it is grounds for divorce.
This law applies even if the husband is not obligated to his wife every seven days. The Mishnah in Kesubos establishes that a camel driver is obligated in the mitzvah of marital relations once in thirty days, and a sailor once in six months. Even so, as long as he has not taken a vow, the way is open to fulfill the mitzvah earlier and the wife hopes for that; but once he has made a formal vow forbidding it, he has driven a wedge between them, and according to Beis Hillel he has only one week to resolve the matter, after which he must give a get.
And the language of the Mishnah: "HaMadir es ishto mitashmish - Beis Shammai omrim: shtei Shabbatos, uVeis Hillel omrim: Shabbas achas" - Beis Shammai hold that the situation may continue for two weeks before it is considered grounds for divorce, and Beis Hillel hold that only one week. And the halachah follows Beis Hillel.
The second dispute: a woman who miscarries on the night of the eighty-first day:
A woman who has given birth is obligated to bring korbanos - a chatas and an olah. Until she brings her korbanos a remnant of tumah remains in her which completes her status as a woman after childbirth, and only after bringing them does she return to eating the meat of kodashim and entering the Beis HaMikdash. If she gave birth to two children and did not bring a korban for the first, she brings two sets of korbanos, one for each child.
A woman who miscarries is likewise obligated in a korban, provided that forty days have passed from conception - this is the measure that obligates a korban. However, if the miscarriage occurred during the days of purity, she is exempt.
She gave birth to a boy: seven days of tumah, after which she immerses and is permitted to her husband; and during the thirty-three days that follow, if she sees uterine blood - it is not impure but pure, and she remains permitted to her husband. In total forty days, at the end of which she brings her korbanos.
She gave birth to a girl: fourteen days of tumah (and from here comes Beis Shammai's proof in the previous dispute), after which there are sixty-six days of purity, during which even if she sees blood she is permitted to her husband. In total eighty days, at the end of which she brings her korbanos and becomes completely pure.
A woman who gave birth to a girl and conceived again within the sixty six days of purity, and then lost the fetus, is exempt from the offering, even if more than forty days of the pregnancy had passed, since a miscarriage that occurs within the days of purity does not count. But if it occurred outside the window of the days of purity, it too obligates her to bring offerings as part of her purification process.
The Mishnah's question: what is the law regarding a woman who miscarries on the night of the eighty first day? The eighty days have already ended, and the following morning she will be eligible to bring her offerings and become completely pure, but offerings are not brought at night. It turns out that the window of the days of purity has already closed, while the window for bringing the offering has not yet opened. Does this miscarriage obligate her in a second set of offerings?
Beis Shammai exempt her: since the window for bringing offerings has not yet opened before her, even though she miscarried on the night of the eighty first she is exempt - and this is the lenient view.
Beis Hillel obligate her: the eighty days were completed and the window of the days of purity closed, and therefore the miscarriage obligates her in a second set of offerings.
And in the words of the Mishnah: "HaMapeles leil shemonim ve'echad - Beis Shammai potrin min hakorban, u'Veis Hillel mechayvin" - a woman who miscarries on the night of the eighty first, Beis Shammai exempt her from the offering, and Beis Hillel obligate her. And the halachah, as one would expect, follows Beis Hillel.
The third dispute: a linen sheet with tzitzis:
The mitzvah of tzitzis actually includes two separate mitzvos: the mitzvah to attach the fringes to the corners of a four cornered garment, and an additional mitzvah that among them there be a thread of techeiles. A clarification is needed here: many people think that techeiles means simply the color blue, as the word means in modern Hebrew, but the intent of the verse and of Chazal is wool dyed blue. There is no techeiles without wool, and so it turns out that woolen threads are attached to the garment.
And here the problem arises: a separate mitzvah in the Torah forbids wearing a garment that combines wool and linen together, the prohibition of shaatnez. How, then, can one attach woolen threads to a linen garment? The answer lies in the juxtaposition of the verses: one verse says "You shall not wear shaatnez, wool and linen together," and immediately afterward it says "You shall make for yourself fringes on the four corners of your garment with which you cover yourself."
This juxtaposition teaches the great principle: when a positive commandment and a prohibition contradict one another, so that it is impossible to fulfill the one without transgressing the other, the positive commandment overrides the prohibition. And it is understood that if there is a way to fulfill both, one is obligated to fulfill both; only when there is no alternative, and we are dealing with a single positive commandment against a single prohibition, does the positive one take precedence. The source of this principle is precisely in the subject before us: the juxtaposition of the mitzvah of tzitzis to the prohibition of shaatnez.
One further introduction: tzitzis is a daytime mitzvah, and the Rishonim disagree whether this means a garment designated for the day or only the daylight hours. A night garment, such as pajamas, is not obligated in tzitzis even if it has four corners; whereas a four cornered day garment worn during the day is obligated.
From here to the complex case: a sheet that people sleep in at night, and which in the days of the Mishnah they would wrap themselves in during the day like a sort of toga. At night it is a bedding item and is not obligated in tzitzis, and during the day it is an actual garment and is obligated. It turns out that if he attached woolen tzitzis to a linen sheet, during the day he fulfills a mitzvah, but at night, once the obligation of tzitzis has lapsed, he is left holding shaatnez and may not sleep in it. The question, then, is: is one obligated to attach tzitzis to a linen sheet that also serves as a garment during the day?
Beis Shammai exempt it: everyone agrees that by Torah law a four cornered garment that is worn is obligated in tzitzis, but Beis Shammai institute a rabbinic decree - similar to the decree not to blow the shofar on Rosh Hashanah that falls on Shabbos, lest one carry it and transgress Shabbos - lest he cover himself with it at night and end up wearing shaatnez and transgressing.
Beis Hillel obligate it: we do not make such a decree. A four cornered garment worn during the day is obligated in tzitzis even though it is linen and the tzitzis are wool, and this is the mitzvah; and a person must be careful not to sleep in it at night.
And in the words of the Mishnah: "Sadin b'tzitzis - Beis Shammai potrin, u'Veis Hillel mechayvin" - a linen sheet with regard to tzitzis, Beis Shammai exempt it, and Beis Hillel obligate it.
The fourth dispute: the Shabbos basket:
The final case deals with produce becoming subject to ma'aser, and the substance of this law is actually taught in the fourth chapter of Maseches Ma'asros, whose main subject is: from when does food become obligated in ma'aser. The principle is that when produce is harvested there is no obligation to separate from it until two conditions are met: gemar melachah - the completion of its processing, and re'iyas penei habayis - bringing it into the house. Therefore someone who picks cucumbers in the field may eat them even without separating, since the conditions have not yet been met.
The Rabbis forbade akilas keva, eating a real meal, from produce that has not been tithed; but akilas arai, casual nibbling, is permitted even in practice from produce that has not reached completion of processing and has not entered the house.
Alongside the Torah obligation there are rabbinic alternatives in which the obligation takes effect even without completion of processing and entry into the house. The Rambam lists five:
He cooked the food.
He sold the food.
He pickled the food. All three are an indication that the produce will no longer reach the usual stages of completion of processing and entry into the house, and therefore the obligation takes effect already now.
He separated Terumah from it - in which case he must separate ma'asros as well.
Shabbos: since eating on Shabbos is a mitzvah, as it says, "and you shall call the Shabbos a delight," every act of eating on Shabbos is considered keva and not arai, and one may not eat produce on Shabbos from which ma'asros have not been separated.
Let us illustrate this: a family that grows cucumbers in their yard and loves pickles has no interest in wasting its crop on eating raw cucumbers, but rather wants to pickle them. As long as they have not been pickled, there is no completion of processing, and therefore the members of the household may eat casually from them at any time. But if they want to eat from them on Shabbos, the obligation to separate takes effect, since Shabbos turns even the eating of cucumbers into a fixed meal.
Hence the question of the Mishnah: what is the law regarding a kalkalah - a basket or vessel designated for Shabbos food - in which one placed foods on Friday for the sake of Shabbos? The members of that family, who do not have time to pickle the cucumbers before Shabbos, select the nicest ones and place them in the vessel designated for Shabbos. May they eat from them on Friday before tithing them? Do we say that once they were designated for Shabbos they have become Shabbos food and require ma'aser, or perhaps as long as Shabbos has not arrived there is no obligation, and one may nibble from them?
"Beis Shammai potrin" - merely placing the food in the vessel does not establish that it will be eaten on Shabbos, and it is still not Shabbos; therefore it is exempt, and one may eat casually from it as long as it has not reached completion of processing.
"Beis Hillel mechayvin" - once it has been placed in the vessel designated for Shabbos it is considered Shabbos food, and the obligation to separate takes effect. Even on Friday one must separate before eating from it, and even for casual eating.
And the language of the Mishnah: "Kalkalas Shabbos - Beis Shammai potrin, u'Beis Hillel mechayvin" - a Shabbos basket - Beis Shammai exempt, and Beis Hillel obligate.
In summary: this Mishnah listed four disputes in which Beis Shammai are lenient and Beis Hillel are stringent: one who vows to forbid marital relations with his wife - two weeks or one week; a woman who miscarries on the night of the eighty-first day - exempt or obligated in a korban; a linen sheet with tzitzis - exempt because of the concern of shaatnez, or obligated; and the Shabbos basket - exempt or obligated in the separation of ma'asros.