Are F and G intersections and unions always σ-fields in set theory?

[Alexsandro]

Can someone help me with this question ?

Let F and G be σ-fields os subsets of S.

(a) Let H = F intersection G be the collection of subsets of S lying in both F and G. Show that H is a σ-field.

(b) Show that F union G, the collection of subsets of S lying in either F or G, is not necessarilt a σ-field.

 

[EnumaElish]

(a) show that the intersection of two closed sets is a closed set (e.g. "closed under complements").

(b) counterexample: Let f be an element of F and g be an element of G. Suppose f U g is neither in F nor in G, therefore not in H. This means H is not closed under union.

P.S. Let S = {1,2,3}, F = {S, ø, {1}, {2,3}}, G = {S, ø, {2}, {1,3}}. Then F U G = {S, ø, {1}, {2,3}, {2}, {1,3}}. If H = F U G was σ, then it would be closed under union, which implies that {1} U {2} = {1,2} would be a distinct element of H. Since it is not, H is not σ.