Discrete Mathematics Answers

If the statement q ∧ r is true, determine all combinations of truth values for p and s such that the statement (q → [¬p ∨ s]) ∧ [¬s → r] is true. (Highlight your final answer not just the truth table)

1-Show that u_x=c₁e^ax +c₂e^-ax solution of the difference equation

u_x+1-2u_xcosha+u_x-1

Construct the conjunctive normal form of the proposition (¬p ∧ q) ∨ (¬r ∨ ¬q).

Construct the disjunctive normal form of the proposition (p → q) ∧ (r ↔ p). 

 If the statement q ∧ r is true, determine all combinations of truth values for p and s such that the statement (q → [¬p ∨ s]) ∧ [¬s → r] is true. 

f p is plotted versus a range of parameter value x the resulting defines ...a?

Let A = {−5, −4, −3, −2, −1, 0, 1, 2, 3, 4}

 and define a relation R on A as follows:

For all x, y  A, x R y ⇔ 3|(x − y).

It is a fact that R is an equivalence relation on A. Use set-roster notation to write the equivalence classes of R.

[0]=  

[1]=  

[2]=  

[3]=  

How many distinct equivalence classes does R have?

List the distinct equivalence classes of R. (Enter your answer as a comma-separated list of sets.)

MATHEMATICAL INDUCTION AND RECURRENC

Solve the following. (10 pts each)

1. Prove P(n) = n2 (n + 1)

2. Recurrence relation an = 2n with the initial term a1 = 2.

 MATHEMATICAL INDUCTION AND RECURRENCE 

5. If P(k) = k2 (k + 2)(k – 1) is true, then what is P (k + 1)? (2 pts)

A. (k + 1)2 (k + 2)(k)

B. (k + 1)2 (k + 2)(k)

C. (k + 1)(k + 3)(k)

D. (k + 1)2 (k + 3)(k)

6. Using the principle of mathematical induction, 2n-1 is divisible by which of

the following? (2 pts)

A. 1

B. 0

C. 4

D. ½ 

7. A relation represents an equation where the next term is dependent on the

previous term is called

A. Binomial relation

B. Recurrence relation

C. Regression relation

D. None of these

8. Calculate the value of a2 for the recurrence relation an=17an-1+30n, where

a0=3. (2 pts)

A. 2346

B. 1296

C. 1437

D. 5484

9. The recurrence relation for Fibonacci sequence is

A. Fn = Fn + 1

+ Fn - 2

B. Fn = Fn - 1

+ Fn - 2

C. Fn = Fn - 1

- Fn - 2

D. None of these

10. In recurrence relation, a0 represents

A. Current value

B. Starting value

C. The value of next term in the sequence

D. None of these 

Solve the following. (mod4)

1. How many 3-digit number can be formed from digits 1 – 5 if:

a. If repetition is allowed?

b. If repetition is not allowed?

2. How many strings of length 4 can be formed using the letters

COMPUTER if repetitions are not allowed?

3. Out of 17 regions in the country, ten will be chosen to be included

in a survey. How many ways of selecting 10 out of 17 regions?

4. Expand the (2𝑥 − 𝑦)3 using binomial coefficient.

5. Find for the coefficient of the following:

a. x4y3 after the expansion of (x – 2y )7

b. x6y6 after the expansion of (2x + y )12

6. Find the 9th term of the expression (2x + y )12