1) WHY IS "ERUSIN" PERMITTED DURING THE WEEK OF TISH'AH B'AV?
QUESTION: Rav Chisda cites the Mishnah in Ta'anis (26b) which states that Erusin may be performed during the week in which Tish'ah b'Av falls. RASHI (DH Mutar l'Ares) explains that "this is because Erusin is not a Simchah."
Rav Chisda then cites a Beraisa which states that before Tish'ah b'Av, people must decrease their activity in business, construction, and agriculture. They may perform Erusin but not Nisu'in or conduct a Se'udas Erusin. RASHI (DH u'Me'arsin) explains that "this is because it is not a Simchah, and because one performs a Mitzvah."
Why does Rashi here add the reason that "one performs a Mitzvah" with Erusin, but he does not mention it in his earlier comment? Moreover, what difference does it make if Erusin is a Mitzvah? Since, as Rashi explains, it is not "Simchah," it should be permitted regardless of whether it is a Mitzvah or not.
ANSWER: Rashi is bothered by the wording of the Beraisa. The Beraisa states that Erusin is permitted, but not Nisu'in or a Se'udas Erusin. If the reason why Erusin is permitted is because no Simchah is involved, then why does the Beraisa need to teach that one may not perform Nisu'in or make a Se'udas Erusin? Those activities obviously are prohibited because they do have Simchah (as the Gemara says on 43b, that it is obvious that Nisu'in is prohibited because it has Simchah)! Why does the Beraisa need to specify that Nisu'in and Se'udas Erusin are prohibited?
It must be that Erusin involves some degree of Simchah. When the Gemara (43b) states that without a Se'udah there is no Simchah with Erusin, it means that Erusin does not involve the same degree of Simchah as Nisu'in (as the TAZ OC 551:3 says). The reason Erusin is permitted, despite the Simchah involved, is that when a person does Erusin he fulfills a Mitzvah.
This explains why the Beraisa must add that Nisu'in and a Se'udas Erusin are prohibited. Since those activities are also Mitzvos, one might have thought that they are also permitted because of the Mitzvah involved, despite the fact that they have Simchah. The Beraisa teaches that since the Simchah of those activities is so great, they are not permitted even though they involve the performance of Mitzvos. This is Rashi's intention when he adds the fact that Erusin is a Mitzvah.

43b----------------------------------------43b

2) THE DIFFERENCE BETWEEN A PERSONAL "AVEILUS" AND A NATIONAL "AVEILUS"
QUESTION: In the Mishnah (41a), Rebbi Yosi permits a woman (who was divorced) to do Erusin with a second husband immediately without having to wait three months. The only exception is a woman whose husband died, in which case the woman must wait before she remarries, not because of the requirement of "Havchanah" but because of her Aveilus.
Rav Chisda and Rava (43a) ask that Erusin for a woman observing Aveilus should be permitted because of a Kal v'Chomer: one is permitted to engage in business during Aveilus but is required to decrease his involvement in business during the days before Tish'ah b'Av. Accordingly, Erusin, which is permitted during the days before Tish'ah b'Av, certainly should be permitted during Aveilus. If the laws of Tish'ah b'Av are more stringent than the laws of Aveilus, Erusin -- which is permitted before Tish'ah b'Av -- certainly should be permitted during Aveilus.
The Gemara concludes that "Aveilus Chadashah differs from Aveilus Yeshanah, and Aveilus d'Rabim differs from Aveilus d'Yachid." RASHI explains that this means that the period before Tish'ah b'Av is more lenient with regard to Erusin because it is a weaker form of Aveilus, since it is an "old" Aveilus (to which people have become accustomed) and it is a "public" Aveilus (and thus people are able to alleviate their grief by sharing it with others).
Although the Gemara's answer explains why Erusin is prohibited during personal Aveilus but permitted during the week of Tish'ah b'Av, it does not explain why business is permitted during Aveilus but prohibited before Tish'ah b'Av. If the Aveilus of Tish'ah b'Av is more lenient as the Gemara assumes, then business -- which is permitted during the more stringent period of personal Aveilus -- should be permitted before Tish'ah b'Av. (TOSFOS DH Sha'ani)
ANSWERS:
(a) RABEINU TAM (cited by Tosfos) answers that the Gemara is not making one statement about the period of Aveilus before Tish'ah b'Av, with two reasons for why that period is lenient. Rather, the Gemara is making two statements about why the laws of Erusin differ from the laws of business. (This understanding is supported by the wording of the Gemara. The Gemara uses the word "Sha'ani" ("it differs") twice, instead of including both differences in one clause.) The first statement refers to Erusin and teaches that since Tish'ah b'Av is an "Aveilus Yeshanah" it is not as stringent as an "Aveilus Chadashah," and thus Erusin is permitted before Tish'ah b'Av. The second statement refers to business and teaches that since Tish'ah b'Av is an "Aveilus d'Rabim" one must be more stringent with regard to acts done in public and acts done for long durations of time. If such acts would be permitted, everyone would see the other person going to work and would assume that work is entirely permitted, and no one would mourn for Yerushalayim. That is why business is prohibited on days of public Aveilus but permitted on days of personal Aveilus. The act of Erusin, in contrast, is not done in public and lasts only for a moment. Permitting Erusin on a day of "Aveilus d'Rabim" will not cause people to neglect mourning for Yerushalayim.
(b) RASHI, however, understands the Gemara differently. He explains that the Aveilus of Tish'ah b'Av is more lenient in two ways -- it is "Yeshanah" and it is "d'Rabim."
According to Rashi, how does the Gemara answer its question of why business is permitted during personal Aveilus but prohibited during the Aveilus of Tish'ah b'Av, if Tish'ah b'Av is a more lenient type of Aveilus?
Perhaps Rashi understands the Gemara in a way similar to that of Rabeinu Tam, that Tish'ah b'Av is more stringent with regard to business, but for a different reason than the one Rabeinu Tam gives. According to Rashi, the very fact that Tish'ah b'Av is not a personal Aveilus requires that efforts be made to help arouse people to mourn. The Chachamim made certain enactments to help people focus on the Aveilus and ponder the loss of Yerushalayim and the Beis ha'Mikdash. They prohibited activities which cause people to take their minds off of mourning, such as business. (Accordingly, the prohibition against business may not be a specific expression of Aveilus at all.) The Chachamim did not prohibit Erusin on such days because Erusin is a momentary act that does not detract from one's focus on mourning for Yerushalayim. In contrast, during an "Aveilus Chadashah," since the Aveilus is personal the Avel will not become distracted from his Aveilus by involvement in business.
(c) The RAMBAN and other Rishonim explain, like Rabeinu Tam, that the two differences -- "Aveilus Yeshanah" and "Aveilus d'Rabim" -- relate to the two different Halachos of Erusin and business activity. However, they give a different logic for why the Gemara is more stringent with regard to business during the days before Tish'ah b'Av than during days of personal Aveilus. They explain that business is prohibited before Tish'ah b'Av not because Tish'ah b'Av is an "Aveilus d'Rabim" but because it is an "Aveilus Yeshanah." Since it is an "Aveilus Yeshanah" people tend to treat it more leniently than a normal Aveilus. In order to ensure that people would take the Aveilus seriously, the Chachamim prohibited business during those days. In contrast, they permitted Erusin because Erusin does not involve Simchah. (This differs from the approach suggested for the words of Rashi on 43a; see previous Insight.) The only reason why the Chachamim prohibited Erusin in a case of normal Aveilus was a Gezeirah lest one perform Nisu'in or lest one make a Se'udas Erusin. During the period before Tish'ah b'Av, since it is an Aveilus d'Rabim no such Gezeirah is necessary; permitting Erusin will not lead to Nisu'in or to making a Se'udas Erusin. This is because the person who performs the Erusin needs others to join him to perform Nisu'in or to make a Se'udah, but they will not join him because they are involved in the Aveilus d'Rabim of Tish'ah b'Av.
(d) The ME'IRI explains, like the Ramban, that "Aveilus d'Rabim" refers to why business is prohibited during the days before Tish'ah b'Av. Since people tend to be lenient on such days, the Chachamim were stringent and prohibited business. He explains, however, that the reason why those days are more lenient with regard to Erusin is that if the Chachamim would have prohibited Erusin during those days, one with a strong desire to betroth a certain woman would ignore the Chachamim. The Chachamim could enact stringent decrees only for activities in which people have no emotional involvement, such as business.
(Tosfos cites RABEINU NISIM GA'ON (in MEGILAS SETARIM) who states that Erusin is permitted not only "Kodem d'Kodem," before the week in which Tish'ah b'Av falls, but it is permitted even during the week in which Tish'ah b'Av falls, and even on Tish'ah b'Av itself. Although both business and laundering clothes are prohibited during that period and are permitted during a normal Aveilus, nevertheless the Chachamim were lenient with Erusin because of the reasons mentioned above: with regard to Erusin, a normal Aveilus is more stringent, and with regard to business or laundering, an Aveilus d'Rabim is more stringent. All of the reasons above apply not only to prohibit business but to prohibit laundering as well.)