Mike Pence took his oath a few minutes before Donald Trump did.
During those few minutes, was he Vice President to President Obama?
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No, because while taking the oath is a prerequisite for executing the office, just swearing it doesn't make one POTUS or VPOTUS.
Article II, Section 1, Clause 8 of the US Constitution says:
Before he [the President] enter on the Execution of his Office, he shall take the following Oath or Affirmation: [...]
"Before" does not necessarily mean "the exact moment" and someone can hold an office without executing it. Term limits are defined in the Twentieth Amendment to the United States Constitution, Section 1, which states:
The terms of the President and Vice President shall end at noon on the 20th day of January
So at January 20th 2017 12:00:00*, the terms of Obama and Biden ended and those of Trump and Pence started. But they still had to take their oaths in order to be allowed to actually do any of those things their offices allow them to do.
* One could nitpick and ask "12:00 according to what time zone?". If you really want to know,
please ask a new question. someone asked that question.