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It seems like fast (with the exception of Tisha B'av and Yom Kippur) is a time bound Mitzva. I exclude Yom Kippur and Tisha B'av because they are all day fasts, so they don't seem to be just time bound Mitzvos.

What makes women obligated to fast on Tzom Gedalia, Asara B'Teves, Taanis Esther, and Shiva Aser B'Tammuz? They are only day time fasts. Why are more obligated in these fasts than say, the Mitzva of Tzitzis? They are both time bound, just day time Mitzvos.

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The release for women from time-bound mitzvos applies only in principle to mitzvos that are positive time-bound mitzvos only.

“Time-bound mitzvos” means any mitzvo that has a specific time (or date) associated with it. So Tisha B'av and Yom Kippur are also time-bound.

Generally the mitzvo not to eat is a negative commandment and so fasts are, in principle, obligatory on women. But women do have reasons to be exempted as is explained in the linked question Why don't Israeli women fast? .

Up to this point was the first version of the answer. Now I add the following:

Fasting on Yom Kippur is a negative mitzvah as stated at www.etzion.org.il

התענית ביום הכיפורים מוגדרת (גם) כמצוות לא תעשה, ולכן נשים חייבות בה

and so women are obliged to fast on it.

Toras Emes states that the reason that women fast on the minor fasts is:

ארבעה צומות חייבים להתענות בי''ז בתמוז בתשעה באב ובג' בתשרי ובעשרה בטבת, מפני הדברים ההרעים שאירעו בהם.

Four fasts: Women are obliged to fast on 17 Tamuz, 9 Av, 3 Tishrei and 10 Teves because of the the bad things that happened on these dates.

The implication from the author giving this reason is that at least he views this reason as being stronger than the reason to fast is a negative mitzvo and possibly he holds that to fast on these days is not a negative mitzvo at all.

Acknowledgement to @mevaqesh for his points.

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    Sources? Particularly for assertion that rabbinic fasts are categorized as "lo ta'aseh" after all, we find biblical prohibitions phrased in the affirmative that are considered positive, e.g. לאו הבא מכלל עשה. – mevaqesh Aug 24 '16 at 14:11
  • A לאו הבא מכלל עשה is defined by Wikiyeshiva as מצווה שכתובה בתורה בלשון עשה, אך קיומה הוא בשב ואל תעשה a mitzvo written in the Torah in a positive form whose fulfilment is through inaction. How is that relevant? (PS I don't have a source yet. ) – Avrohom Yitzchok Aug 24 '16 at 16:35
  • Because the fact that a mitzvah involves inaction Is not proof that it Is classified as a negative commandant (in which women are obligated) – mevaqesh Aug 24 '16 at 19:29
  • @mevaqesh Thank you very much. I have updated the answer in light of your excellent comments. – Avrohom Yitzchok Aug 24 '16 at 20:47
  • I note the Chinukh 313 counts fasting as a positive command (תענו את נפשותיכם) but later in 316 counts "not eating" as a negative command. See too the Rambam's listing in the introduction to his laws of rest on YK mechon-mamre.org/i/3300.htm. Also we know it gets Kareit and isn't listed among the positive commandments that get Kareit (Kereitot 1:1). – Double AA Aug 24 '16 at 21:00

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