The math club at Podunk University has a number of committees according to the following rules established in its constitution: (1) There must be at least one committee. (2) Each committee has at...

The math club at Podunk University has a number of committees according to the following rules established in its constitution: (1) There must be at least one committee. (2) Each committee has at least three members. (3) Any two distinct committees have exactly one member in common. (4) For each pair of members of the club, there is one and only one committee of which both are members. (5) Given any committee, there exists at least one member of the club who is not a member of that committee. Prove the following.

(a) There exist at least three members in the club.

(b) There exist at least three committees.

(c) If x is a member of the club, then there is at least one committee that does not have x as a member.

(d) Every member of the club is a member of at least three committees.

(e) There are at least seven members in the club.

(f ) There are at least seven committees.

(g) Describe a club with seven members and seven committees that meets all the required conditions.