discrete structures

Let A = {1, 2, 3, 4, 5, 6, 12} and define the relation R on A by m R n iff m|n. Write the definitions of the properties, reflexive, antisymmetric and transitive and the use the definitions to determine whether each property holds for this relation. (a) Is this relation a partial ordering relation? Why? If so, draw its Hasse diagram. (b)Write the (boolean, that is, the yes/no) matrix of this relation. I know the definitions but am having difficulty in determining the usage for each. I'm really stuck on this one.

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

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

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!