Find the cardinality of C which is smallest -algebra generated by the collection

Question

Find the cardinality of C which is smallest -algebra generated by the collection of X = {a,b,c,d}, {a}, {b}, {c}. If we replace -algebra by algebra, are the two collections’ cardinalities equal? - please show the step clearly. I will rate you if you get it right! Find the cardinality of C which is smallest -algebra generated by the collection of X = {a,b,c,d}, {a}, {b}, {c}. If we replace -algebra by algebra, are the two collections’ cardinalities equal? - please show the step clearly. I will rate you if you get it right!

Explanation / Answer

An algebra is a collection of subsets closed under finite unions and intersections and A sigma algebra is a collection closed under countable unions and intersections.

and in finite case we can conclude by definitions that algebra and sigma algebra both are same as countable becomes finite in case of finite set ---------*

take finite unions of {a,b,c,d} ,{a},{b},{c}

{a,b,c,d},{a,b},{a,c},{b,c}

finite intersectio --- phi, {b,c,d},{a,c,d},{a,b,d}

hence algebra is {{a,b,c,d},{a,b},{a,c},{b,c}, phi, {b,c,d},{a,c,d},{a,b,d}}

hence cardinality is 8 and by argument * algebra and sigma algebra are same hence cardinalities are same and equal to 8.

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