Note: I replaced #(1rarr#~#r)# in the question with #(qrarr#~#r)#
since it seemed the only logical place to fit in the #q# that appears in all the answer choices.
negation of #p^^(qrarr#~#r)#
#color(white)("XXX")-=color(red)( #~#(p^^(qrarr#~#r)))#
In the more general case:
#color(white)("XXX")#~#(p^^s) -= #~#p vv #~#s#
So (replacing #s# with #(qrarr#~#r)#
#color(white)("XXX")color(red)(#~#(p^^(qrarr#~#r)))-=color(blue)(#~#pvv#~#(qrarr#~#r))#
Also in the general case:
#color(white)("XXX")qrarrt -= #~#qvvt#
#color(white)("XXX")#~#(qrarrt)-= #~#(#~#qvvt)-=q^^#~#t#
So
#color(white)("XXX")color(blue)(#~#pvv#~#(qrarr#~#r)) -= color(green)(#~#pvv(q^^r))#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
If the original version of the question was correct (but the answer choices were wrong):
#1-=true#
So
#color(white)("XXX")(1rarr~r)-= ~r#
and
#color(white)("XXX")~(p^^~r)-=~pvvr#
