Discrete Mathematics Answers

State the Pigeonhole Principle. In a result sheet of a list of 60 students, each marked “Pass” or “Fail

“. There are 35 students pass. Show that there are at least two students pass in the list exactly nine

students apart. (for example students at numbered 2 and 11 or at numbered 50 and 59 satisfy the

condition).

State the Pigeonhole Principle. A chess player wants to prepare for a championship match by playing

some practice games in 77 days. She wants to play at least one game a day but no more than 132

games altogether. Prove that there is a period of consecutive days within which she plays exactly 21

games

List the members of these sets

1.    Let X be the set of positive integers less than 10

2.    Let Y be the set of negative even integers > -10

3.    Let Z be the set of positive odd integers < 10

Answer what is asked.

Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

A = {1, 2, 4, 6, 8, 10}

B = {1, 3, 5, 7, 8, 9}

C = {2, 4, 6, 8, 10}

D = {4, 5, 6, 7, 8, 9, 10}

Find?

1.  A’ =

2.  B’ =

3.  C’ =

4.  D’ =

5.  A' ∪ B' =

6.  C' ∩ D' =

7. (A ∩ B) U C =

8.  A ∩ (B U D) =

9. (A ∩ D) - C =

10.(D' ∩  A') - B = 

Consider a recurrence relation a_n = -3a_n-1 + a_n-2 for n = 1,2,3,4,… with initial conditions a₁ = 4 and a₂ = 2. Calculate a₅.

Consider a recurrence relation a_n = 3a_n-1 for n = 1,2,3,4,... and suppose a₁ =4. Calculate a₅.

What and Define Equivalences

What and Define implications

What and define Rules of inferences

What and Define Equivalent formula or laws of algebra of propositions