The negation of the statement

“Rajesh or Rajni lived in Bangalore” is

If the statement is p then its negation is ~p, it means if p is true then ~p is false and vice versa.

p: Rajesh or Rajni lived in Bangalore

q: Rajesh lived in Bangalore

~q: Rajesh did not lived in Bangalore

r: Rajni lived in Bangalore

~r: Rajni did not lived in Bangalore.

Now,

~(qVr)=~q ᴧ ~r

Hence, Rajesh did not live in Bangalore and Rajni did n ot live in Bangalore.