Please help with problem 7. References are below the image.

7. 14 points Let @ be a nonempty collection of subsets of a nonempty set X and a subset B C X be arbitrary. Prove that (a) r(6n B) = (6)n B, where 60B := (COB|CEC); (b) or(6 n B) = or(6) n B.Definition 2.6 (Monotone Class). Let A be a nonempty set. A monotone class (of sets) on X is a nonempty collection # of subsets of X such that, if (An )men is a monotone sequence in /, then lim An E # ( closedness under the limits of monotone set sequences). n-+00Definition 2.2 (Ring, Algebra). Let X be a nonempty set. A ring (of sets) on X is a nonempty collection ? of subsets of X such that (1) 0 EX; (2) if Al, . .. . An ES (n E N), then JA e (closedness under finite unions); =1 (3) if A, BE , then A \\ B E 2 ( closedness under set differences). An algebra of (of sets) on X is a ring on X such that X e . Definition 2.4 (0-Ring, o-Algebra). Let X be a nonempty set. A o-ring (of sets) on X is a nonempty collection of subsets of X such that 1) DER; (2) if (An )nen is a sequence in 2, then A E R ( closedness under countable i=1 unions); 3) if A, BE , then A \\ B E (closedness under set differences). A o-algebra _ (of sets) on X is a o-ring on X such that X E E. Definition 2.7 (Generated Set Classes). Let & be a nonempty collection of subsets of a nonempty set X. O r(8) := n is the smallest ring containing 4 called the ring is a ring, "CQ generated by C. - a(6) := n of is the smallest algebra containing & called the of is an algebra, "EC a algebra generated by C. or( 8 ) : = n is the smallest o-ring containing & called the of is a o-ring, CCs o-ring generated by . go() := n E is the smallest o-algebra containing & called E is a o-algebra, CCE the a-algebra generated by C. m() := n # is the smallest monotone class contain- W is a monotone class, CC.# ing & called the monotone class generated by C