Question

Problem 5. (8 points) Consider rolling a six-sided die. Let A be the set of outcomes where the roll is an even number Let B be the set of outcomes where the roll is greater than 3. Calculate the sets on both sides of De Morgan's laws and verify that the equality holds. (AUB)c A n B

Answer 1

Using definitions of A and B we have

A={2,4,6} , B={4,5,6} and $A^c=\{1,3,5\}~,~B^c=\{1,2,3\}$

So, $A\cup B=\{2, 4,5,6\}$ ​​​​​​, and $A^c\cap B^c=\{1,3\}~,~(A\cup B) ^c=\{1,3\}$ ​​​​​​

Therefore, $(A\cup B) ^c=A^c\cap B^c$ ​​​​​​

Also, $A\cap B=\{4,6\}$ ​​​​​​, and  $(A\cap B) ^c=\{1,2,3, 5\} ~~,~~A^c\cup B^c=\{1,2,3,5\}$

Hence ​$(A\cap B) ^c=A^c\cup B^c$