Discrete Mathematics Answers

Find out if the following functions are one-to-one and/or onto. a. 𝑓: 𝑍 → 𝑅, 𝑓(𝑥) = 𝑥 3 + 1 b. 𝑓: 𝑅 + → 𝑅 +, 𝑓(𝑥) = |𝑥| + 5

Let  . P(x): x²/2 = x

Find the following then identify their truth values.

1. P (1)
2. P (2)
3. ꓯn, P(n)
4. ⱻn, P(n)

4. For each of these pairs of sets, determine whether the first is a subset of the second, the second is a subset of the first, or neither is a subset of the other. a) the set of people who speak English, the set of people who speak English with an Australian accent b) the set of fruits, the set of citrus fruits c) the set of students studying discrete mathematics, the set of students studying data structures

What is the probability that when a coin is flipped six times in a row, it lands heads up every time?

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)