Discrete Mathematics Answers

Given the recurrence relation 𝑎𝑛 =−2𝑎𝑛−1 +15𝑎𝑛−2 with the initial conditions 𝑎0=1 and 𝑎1=7.

(a) Write the characteristic equation.

(b) Solve the recurrence relation.

P(x): x²/2 = x

Find the following then identify their truth values.

1. P (1)
2. P (2)
3. An, P(n)
4. En, P(n)

Use quantifiers to express the Statement below.

• Let P(x,y) be the statement. " x loves y" where D for both x and y is the set of all people in the world

• Denote the propositions given in the next slide

- Everyone loves everyone.

- For everyone, everyone loves them.

- Everyone loves someone.

- There is someone who loved by everyone.

- There is someone who loves everyone.

- For everyone, there is someone who loves them.

- There is someone who loves someone

- There is someone who is loved by someone.

Let P(x,y) be the statement "student x has taken a class y", where the domain for x consist of all students and y consist of all computer engineering courses/subject at your school .

- Expree each of the quantification in

English sentences.

• ∀x∀yP(x,y)

• ∀x∃yP(x,y)

• ∃x∀yP(x,y)

• ∃x∃yP(x,y)

1, In how many ways each 5 boys and 3 girls be seated around a table if:

i) There is no restriction?

ii, Boy Bi and girl Gi are not adjacent?

ji, No girls are adjacent?

2. Each user on a computer system has a password, which is six to eight characters

long. where each character is an upper case letter or a digit Fach password must

contain at least one digit. How many possible passwords are there?

3.how many ways are there to distribute 10 identical bones to 4 dogs if each dog must

stat least | bone and Fifi may not receive more than 3 bones’?

1.     True or false: (~p∧q) ∨ (~p∨q)≡ (~p∨q)

1.4 If Universal Set U = {90, 91 , 92 , 93 , 94, 95 , 96 , 97 , 98, 99 , 100} (10) A = {90, 92, 94, 96, 98, 100}, B= {91, 93, 95, 97, 99}, C = {90, 94, 98}

1.4.1 What is (A ∩ C)c

1.4.2 What is (B ∪ C)c

Find the inverse of the encrypting function

F(x) =(5-x) mod 26 , 0<=x<=25

And use it to decrypt the massage '' ZRQUBFNB''