Discrete Mathematics Answers

There are 6 people to be arranged in a line for a concert. How many

arrangements are possible?

Prove by PMI for 2^n+1>2n+1

Two functions f : R → R and g : R → R are defined by f(x) = 5x3 + 1 and g(x) = 2x − 3 for all x ∈ R.

Determine the inverse of (f -1 ◦ g) and (g ◦ f )(2) and ( f ◦ g)(2).

Solve the inequalities. Give your answer in interval notation, and indicate the answer

geometrically on the real-number line.

a. t + 6 ≤ 2 + 3t

b. 3(2 – 3x) > 4(1 – 4x)

Let the function f : R → R and g : R → R be defined by f(x) 2x + 3 and g(x) = -3x + 5.

a. Show that f is one-to-one and onto.

b. Show that g is one-to-one and onto.

c. Determine the composition function g o f

d. Determine the inverse functions f -1 and g -1 .

e. Determine the inverse function (g o f) -1 of g o f and the composite f -1 o g -1 .

Determine whether each of the following functions from Z to Z is one to one and onto.

a.f(n)=n-1

b.f(n)=[n/2]

C.f(n)=n²+1

a) A three digit number is to be formed using the digits 1, 2, 3, 4, 5, 6 and no repetition is

allowed.

i) How many numbers can be formed if the leading digit is 4?

ii) How many numbers can be formed if the number is more than 250?

iii) How many odd numbers can be formed between 200 and 400?

b) Consider a bookshelf contains 28 books in different genre. 14 books are in education, 9

books in business and 5 books in motivation. A student would like to take 15 books. Find the number of ways if:

i) there is no restriction

ii) the choice must consist of 8 books in education, 5 books in business and 2 books in

motivation genre.

iii) The choice must consist of at least 9 books in education and exactly 5 books in

motivation genre.

Two functions f : R → R and g : R → R are defined by f(x) = 5x3 + 1 and g(x) = 2x − 3 for all x ∈ R.

Determine the inverse of (f -1 ◦ g) and (g ◦ f )(2) and ( f ◦ g)(2).

Let the function f : R → R and g : R → R be defined by f(x) 2x + 3 and g(x) = -3x + 5.

a. Show that f is one-to-one and onto.

b. Show that g is one-to-one and onto.

c. Determine the composition function g o f

d. Determine the inverse functions f -1 and g -1 .

e. Determine the inverse function (g o f) -1 of g o f and the composite f -1 o g -1 . 

How many ways are there to select 12 countries in the United Nations to serve on a council if 2 is selected from a block of 55, 2 are selected from a block of 67 and 8 are selected from the remaining 67 countries