if and are sets, then we define the cartesian product to be the collection of ordered pairs, whose first component lies in and second component lies in , thus or equivalently
some stuff from college
cartesian product of 2 sets is the set of all the possible ordered pairs that can be obtained by taking an element from as the first in the pair and an element from as the second
assume
the cartesian product of where is:
assume
power:
or or
assume or or need to prove:
we split into cases:
case 1: assume
case 2: assume , same as
case 3: assume
assume need to prove: or or
we assume in contradiction that and and exists an so that and or there exists an so that and we split into cases:
case 1: there exists an so that and exists so that and there exists an so that and there exists so that and case 2: there exists so that and , a similar path to the previous case
we arrived at a contradiction so the theorem is true