how do i open show $(A_1 \cup A_2) -(B_1 \cap B_2)=(A_1-B_1)\cup(A_1-B_2)\cup(A_2-B_1)\cup(A_2-B_2)$?

Question

I'm having problem with proving this following relation:

$(A_1 \cup A_2)-(B_1 \cap B_2)=(A_1-B_1)\cup(A_1-B_2)\cup(A_2-B_1)\cup(A_2-B_2)$

i understand the logic behind it, but when i to open $(A_1 \cup A_2)-(B_1\cap B_2)$ i don't get to the final equitation.

i tried to use elementary identities such as $a-b=a\cap b'$ but still couldn't make it.

thank you for helping

Answer 1

1. Using the elementary identity you wrote down ($a-b=a\cap b'$) gives you

$$(A_1\cup A_2)-(B_1\cap B_2) = (A_1\cup A_2)\cap((B_1\cap B_2)')$$

2. Now use De Morgan's law to change $(B_1\cap B_2)'$ into an expression involving unions.

3. Now use the distributive law