table of contents

cartesian product

2024-01-01

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