Discrete Mathematics Answers

36. Show that the propositions pi, p2, p3, and p4 can be shown to be equivalent by showing that p1 ↔ p4, P2 ↔ P3, and p1 ↔ p3.

the number of combinations of five objects taken two at a time is?

Let ρ be a relation on a set A. Define ρ −1 = { | ∈ ρ}. Also for two relations ρ, σ on A, define the composite relation ρ ◦ σ as (a, c) ∈ ρ ◦ σ if and only if there exists b ∈ A such that ∈ ρ and ∈ σ. Prove the following assertions (a) ρ is both symmetric and antisymmetric if and only if ρ ⊆ { | a ∈ A}. (b) ρ is transitive if and only if ρ ◦ ρ = ρ. 

For discrete structures there are n exams to check and there are k graders. To guarantee a high quality of grading every exam may be checked by any number of graders (but always at least by one grader). This means that summed all together the graders may make up to k ∗ n exam checks. To avoid this it is required that for each pair of graders there is at most 1 exam that they have both checked. Prove that this rule creates a much better bound of at most ((k +n) 3/2 + (k +n))/2 exam checks. Hint: Consider modeling the grading work as a graph. 

Find the sum of product expansion of the Boolean function f(x,y,z) = (x+z)y