Question

How do you prove this?
(A U B) ∩ (A' U B') = (A ∩ B') U (A' ∩ B)

Do you use the complement law in proving this or the distributive property?

Answer 1

hey let me look at this

Answer 2

i assume B' means not B (or the complement)

Answer 3

yes. It's B complement

Answer 4

we need to show they are both subsets of each other

Answer 5

how do I show it?

Answer 6

do I simply write (A U B) ∩ (A' U B) ⊂ (A ∩ B') U (A' ∩ B)?

Answer 7

yes we need to show that

Answer 8

let x \[\in\] (A U B) n (A' U B')
=> x \[in\] A U B and x \[in\] A' U B'
do you agree so far?

Answer 9

oops that didn't come out the way i thought it would

Answer 10

will using the distributive property help in proving that they are both subsets of each other?

Answer 11

x in A U B => x is in A or x is in B or x is in both A and B
x in A' U B'=>x is in A' or x is in B' or x is in both A' and B'
If we Assume x is is in both A and B, then x is not in A' and x is not in B'
so x is not in both A and B
and
x is not in both A' and B'
if we assume x is in A, then x is not in A'=> x is in B'
so x in A n B'
so x in (A n B') U (A' n B)
if we assume x is in B, then x is not in B'=> x is in A'
so x in A' n B
so x in (A n B') U (A' n B)

Answer 12

so (A U B) n (A' U B') C (A n B') U (A' n B)

Answer 13

I'm kind of getting it already. Thanks for your help

Answer 14

k