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I know that many commandments, e.g. keeping Shabbat, not lying, etc., must be violated if doing so is necessary to save a life.
For which commandments is this not true? Does it ever matter how many lives it saves (for example, could you kill one person to save the lives of five), or whether or not it is your own life?
See Rambam Hilchot Yesodei Hatorah Chapter 5
Basically:
The Torah says (Vayikra 18:5) "which a man will perform and live by them". Our Rabbis learn that this means a person doesn't have to kill himself to perform Mitzvot. The exception to this rule are "The Big Three": idol worship, forbidden sexual relations, and murder. If someone gives him a choice to transgress these or die, he must choose to die. There are however, caveats, this does not apply across the board.
With Regard to other Mitzvot (not the Big Three):
Important note: The Rambam is of the opinion that a person may not sacrifice his life for any Mitzvah that is not the Big Three. The Kesef Mishna (commentary on the Rambam) says that many Poskim say that one can choose to sacrifice himself, even if he doesn't have to Halachically.
In a regular time (i.e. when there is no decree against Jewish practice)
If the Jew is being told to transgress the commandments for the non-Jew's benefit (i.e. selfish reasons), he can transgress the commandment.
If the Jew is being told to transgress the commandment, because he wants the Jew to go against G-d's Torah (i.e. ideological reasons), it depends:
if it is in public (i.e. ten Jewish witnesses), he must give up his life
if it is in private, he should transgress
According to the Nimukei Yosef (brought in the aforementioned Kesef Mishna), even the Rambam would agree that a great and holy man may choose to give up his life for any Mitzvah, in order to make a point to the Jews of that generation about the holiness of all the precepts in the Torah.
In a time of decree against Jewish practice
One must give up his life for any commandment, whether it is public or private.
The Talmud (Sanhedrin 74A see English here) adds that even if the decree is against a Jewish practice that is not a Mitzvah (e.g. changing the color of your shoe strap to conform to non-Jewish styles), one must give up their life rather than comply.
With regards to giving up one person to save many, the Rambam (Halacha 5) generally prohibits it unless the Jews are asked to give up a specific person who was already sentenced to die by the hands of a Jewish court.
It does not matter how many lives you would be saving by giving up the one person, you are not allowed to do it.
And see Halachot 6-8 for which prohibitions are allowed to be transgressed in order to save someones life.
they are 3, avoda zara, forbidden relations and murdering.
