Question #10ec0

1 Answer
May 18, 2016

negation of #p^^(qrarr#~#r)# is equivalent to #(4) color(green)(#~#pvv(q^^r))#

Explanation:

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#