The math club at Popular College has a number of committees according to the following rules established in its constitution: (1) Every member of the club is a member of at least one committee. (2)...

The math club at Popular College has a number of committees according to the following rules established in its constitution: (1) Every member of the club is a member of at least one committee. (2) For each pair of members of the club, there is one and only one committee of which both are members. (3) For every committee there is one and only one committee that has no members in common with it. (4) Every committee must contain at least one member of the club. Prove the following.

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

(b) If the club has at least one member, then it has at least four members.

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