1. Mustahhab (recommended) fasts are NOT permissible while traveling. So one cannot fast on the day of Arafah if he is traveling.
2. Yes. For Nazr, there are 3 scenarios:
a. A person does a Nazr for a Mustahhab fast and doesn't specify a day: He can't fast while traveling
b. A person does Nazr that he will fast on a particular day during a journey: He should observe this fast while traveling
c. A person does Nazr that he will fast on a particular day (for instance: the day of Arafah): He should fast disregarding he is traveling or not.
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