Discrete Mathematics Answers

What and Define The Hamming Metric with example in mathematical foundations of computer science

Show that the relation R = ∅ on a nonempty set S is symmetric and transitive but not reflexive.

Answer the following items. Show your complete answer on a separate sheet of paper.

Prove that the following sentences are tautologies.

1. p →p

2. p → (p V q)

3. [p Λ (p → q)] → q

4. p V ~p

5. q → (p V ~p)

6. ~p → (p →q)

7. (p Λ q) → p

8. (p → q) → [(p V r) → (q V r)]

9. ~q → ~(q Λ r)

Let R={(1,2),(1,4),(2,1),(2,4),(3,2),(3,4)}

R={(1,2),(1,4),(2,1),(2,4),(3,2),(3,4)}

is a relation on set A={1,2,3,4}

A={1,2,3,4}

Suppose a Rn b

means that there is a path of length n from a

to b

Which of the elements are R3?

How many pairs of dance partners can be selected from a group of 12 women

and 20 men?

4. Let A and B be sets. Prove the commutative laws from Table 1 by showing that

a) A ∪ B = B ∪ A.

b) A ∩ B = B ∩ A.

a) In how many ways can a committee of 5 be chosen from 9 people?

(b) How many committees of 5 or more can be chosen from 9 people?

(c) In how many ways can a committee of 5 teachers and 4 students be chosen

from 9 teachers and 15 students?

(d) In how many ways can the committee in (C) be formed if teacher A refuses

to serve if student B is on the committee?

bit is either 0 or 1: a byte is a sequence of 8 bits. Find

(a) the number of bytes that can be formed

(b) the number of bytes that begin with 11 and end with 11

(c) the number of bytes that begin with 11 and do not end with 11 and

(d) the number of bytes that begin with 11 or end with 11.

Find the expansion of

a) (x + y)^6

b) (x + y)^4

31) What is the coefficient of x^12 y^13 in the expansion of (x + y)^25?

32) What is the coefficient of x9 in (2 − x)19?

33) What is the coefficient of x^101 y^99 in the expansion of (2x − 3y)^200?

Use the binomial theorem to find the coefficient of x^a y^b in the expansion of (2x^3

− 4y^2)

7, where

a) a = 9, b = 8.

b) a = 8, b = 9.

c) a = 0, b = 14.

d) a = 12, b = 6.

e) a = 18, b = 2.

35) How many unique partitions of the word ARKANSAS are there?

36) A group of six students consists of 3 seniors, 2 juniors, and 1 sophomore. How

many unique partitions of this group of students are there by grade?