Hasse diagram

I am just uncertain if I understand his properly - here is the question and my answer. please feel free to correct me. Consider the following Hasse diagram of a partial ordering relation R on a set A: (a) List the ordered pairs that belong to the relation. My ANS: (a,a),(b,a)(b,b),(c,a),(c,b),(c,c),(d,a),(d,b)(d,d) (b) Find the (boolean) matrix of the relation. < my answer file attached as Mr.jpg> I did review another answer posted along the same question, but uses a different shape. I think my matrix is correct based on my pairs, but i'm concerned about the pairs.

Discrete mathematics Growth Functions big oh big theta

(See attached file for full problem description with proper symbols) --- 1.The functions f and g from the set of real numbers to the set of real numbers are asymptotic or f~g if lim x∞ f(x)/g(x)=1. Let f(x) = log (1+x2) and g(x) = log x. Prove that f(x) is g(x)) but fand g are not asymptotic.

Euler's theorem applied to large integers

Consider Euler’s theorem: If m is a positive integer and a is an integer relatively prime to m, then a^phi(m)≡1(mod m) Use this theorem to show that if a is an integer relatively prime to 32760 then a^12≡1(mod 32760). Symbols better shown in file (attached).

Proofs divisibility

Let n be a positive integer a) prove that n is divisible by 5 if and only if it ends with 0,5 b) prove than n is divisible by 11 if and only if the alternating sum of its digits is divisible by 11 c) find a similar criterion for divisibility by 7 and prove it .

Normal distribution

Please see the attached file "Q11.5.doc" for the problem statement. suppose that w is normal and has expected value a and variance sigma^2. so the expected value of u = exp(w) is exp(a + 0.5sigma^2). What is the variance of u: var(u)?