Check all the statements that are true The power set of the

Check all the statements that are true: The power set of the empty set is empty. The range of a function is always a subset of its codomain. It is possible for a set and its complement to both be empty. If g: A rightarrow B, f: B rightarrow C are injective functions, so is f g. The complement of the intersection of two sets is the intersection of their complements. The sum of the first n positive integers is n (n + 1)/2. If g: A rightarrow B, f: B rightarrow C are functions and f g is a bijection, then f and g must also be bijective. The power set of a set always has double the cardinality of the set. The sum of the squares of the first n positive integers is n (n + 1)(n + 2)/3

Solution

A) false as a power set is never empty . Power set of empty set contains one element.

B) true as function maps values of domain to its codomain . So range is a subset of codomain , may or may nit be equal.

C) no a set and its complement are never empty together.

D) true as if f and g are injectives then fog is injective too.

E) false as De M9rgan law says complement of intersection of two sets is union of their complement.

F) true as by arithmetic progression sum formula we can have it .

G) false if fog is bijective then f and g may not be bijective.

H) false as power set of any set has cardinality 2 raise to the cardinality of the ser

I) false as it is n(n+1)(2n+1)/6