Discrete Mathematics Answers

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,...}

How many permutations of the 26 letters of our alphabet do not contain any of the three strings "US", "AIM", and "DONKEY"?

Let X be a non-empty set, and let R be an equivalence relation on X. Let C be the set of all equivalence classes of R. So C={A⊆X such that A=[x] for some x ∈ X}.

Now, define f : X → C by the rule f(x) = [x] for all x ∈ X.

Prove that if x ∈ X, then there is one and only one equivalence class which contains x.

Suppose X = {1, 2, 3, 4, 5} and that R is an equivalence relation for which 1 R 3, 2 R 4 but 1 R̸ 2,1 R̸ 5,and 2 R̸ 5.

Write down the equivalence classes of R and draw a diagram to represent the function f.

Let X = {1,2,3}. Define a relation ∼ on P(X) by A ∼ B if A and B have the same number of elements.

Prove that ∼ is an equivalence relation and write down all equivalence classes of ∼.