Find the cardinality of C which is smallest algebra generate

Solution

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.