transitive closures

I've attached the problem I'm having trouble with. I put the example I'm trying to work with in pink font. Please help! --- (See attached file for full problem description)

Word problem - A person has 14 close friends.

A person has 14 close friends. (a) Suppose that two of her friends (of the 14 of either gender) do not like each other. If one of the two is invited, the other will not come to the party. How many ways are there to invite 8 people to the party. Explain. (b) Suppose that two of her friends are a couple. She cannot invite one of the 2 without inviting the other. How many ways are there to invite 8 people to the party. Explain.

Prove theta Relation : Reflexive, Symmetric and Transitive

Prove that theta is a reflexive, symmetric, and transitive relation; that is for all f, g, h: N to N, a. f belongs to theta f; b. f belongs to theta g then g belongs to theta f; c. f belongs to theta g and g belongs to theta h then f belongs to theta h;

Logarithmic Proof : ( x^log y = y ^ log x )

From the definition of log, prove that: x to the log y power is equal to y to the log x power; ( x^log y = y ^ log x ) Show all work!

Golf Scheduling Problem : Combinations and Groups

I have 12 golfers total, and we are playing 6 rounds of golf with 4 golfers in each group. I want to try and have everybody play with each person at least once.