Discrete Mathematics Answers

Define the relation R as
R = {(a, b)|a, b ∈ Z, 4 divides a − b}. Show that R is reflexive, transitive and symmetric.

Prove that 3n2 − n + 10 is an even integer for all integers n. (Hint: Prove the statement
true when n is odd, then prove it is true when n is even.)

Write down the contrapositive of the statement “If 3n2 + 4 is even then n is even.” Then prove the statement for all integers n.

Prove that
A − (B ∪ C) = (A − B) ∩ (A − C),

Prove that for all sets A and B,
A ∪ B^c = B^c ∪ (A ∩ B).

Prove that
A − (B ∪ C) = (A − B) ∩ (A − C),
for sets A, B and C.

Prove that for all sets A and B:
A⊆B ⇐⇒ A∪B=B.

How many ways can you assign 4 caretakers each of which look after 2 lighthouses for a year (for a total of 8 lighthouses), and then assign each two lighthouses in the next year so that none of them tend the same two lighthouses both years?

How many ways can you assign 4 caretakers each of which look after 2 lighthouses for a year (for a total of 8 lighthouses), and then assign each two lighthouses in the next year so that none of them tend the same two lighthouses both years?

Find a generating function in closed form for the sequence:
{1,2,3,4,1,2,3,4,1,2,3,4,...}