I'm applying to grad schools in the U.S. for a Ph.D. in mathematics and I recently read about the key differences between an SOP(Statement of Purpose) and a PS(Personal Statement) after noticing a few universities requiring both separately. My question is for the universities that only ask for one statement and/or use the two words interchangeably, can I just add the two statements one after the other to create a single pdf file? While I wish to keep the language in the sop more formal and explain what my current and future plans are with this degree and why I'd make a good researcher, I'll be using the PS to be slightly less formal and talk more about my life experiences and the role they've played in developing my interests in my field as well as explain the gaps and shortcomings in my CV.
Ideally, I would like to keep these separate but for the universities that require only a single SOP, should I expect that they're just not interested in my PS or that they are using the words interchangeably and they want me to combine details form both letters into a single one, instead of appending two letters together?