PART ONE
a. Factorize: px - 2qx - 4qy + 2py
b. Given that the universal set U = {1,2,3,4,5,6,7,8,9,10}, P = {1,2,4,6,10} and Q = {2,3,6,9}. Show that (P ∪ Q)\(^I\) = P\(^I\) ∩ Q\(^I\).
Verified Correct Answer: .
Step-by-Step Mathematical Solution
a. px - 2qx - 4qy + 2py = x(p - 2q) - 2y(2q - p) = x(p - 2q) + 2y(p - 2q) = (x + 2y)(p - 2q)
b. U = {1,2,3,4,5,6,7,8,9,10}
P = {1,2,4,6,10}
Q = {2,3,6,9}
(P ∪ Q) = {1,2,3,4,6,9,10}
(P ∪ Q)\(^I\) = Elements in U but not in (P ∪ Q)
(P ∪ Q)\(^I\) = {5,7,8}
P\(^I\) = {3,5,7,8,9}
Q\(^I\) = {1,4,5,7,8,10}
P\(^I\) ∩ Q\(^I\) = {5,7,8}
Evidently, (P ∪ Q)\(^I\) = P\(^I\) ∩ Q\(^I\)
Source: Myschool Past Questions Archive (Record #75919)