Discrete Mathematics Answers

Use any of the two proof methods to prove:

((~a^b)^(b^c))^~b

Exercise 9:

Draw a full binary tree having the following properties

1. Four internal vertices and five terminal vertices.

2. Height = 3 and nine terminal vertices.

3. Height = 4 and nine terminal vertices.

a)     Suppose P (x, y) denotes the equation8, what will the truth values of the Propositions P (2, 2), P (0, 4)?

In how many ways can 10 people be seated in a row so that a certain pair of

them are not next to each other?

Three persons enter into car, where there are 5 seats. In how many ways can

they take up their seats?

How many numbers can be formed using the digits 1, 3, 4, 5, 6, 8 and 9 if no

repetition is allowed?

13) Suppose that we draw a card from a deck of 52 cards and replace it before the

next draw. In how many ways can 10 cards be drawn so that the tenth card is a

repetition of a previous draw?

How many different plates are there that involve 1, 2 or 3 letters followed by 1,

2, 3 or 4 digits?

How many 2 digit or 3-digit numbers can be formed using the digits 1, 3, 4, 5, 6,

8 and 9 if no repetition is allowed?

If 2 distinguishable dice are rolled, in how many ways can they fall? If 5

distinguishable dice are rolled, how many possible outcomes are there? How

many if 100 distinguishable dice are tossed?

The chairs of an auditorium are to be labeled with a letter and a positive integer

not exceeding 100. What is the largest number of chairs that can be labeled

differently?

Let f : A → B, g : B → C, and h : C → D be functions.

1. State what you need to show to conclude that h ◦ (g ◦ f) = (h ◦ g) ◦ f. 13

2. Consider now some a ∈ A. Calculate h((g ◦ f)(a)) and (h ◦ g)(f(a)). Are they equal?

3. Use your solutions to (1)–(2) to conclude that h ◦ (g ◦ f) = (h ◦ g) ◦ f.

Find a relation R such that 𝑥+𝑦 2 >1 if A = {0,1, 2} and B ={0, 1, 2, 3}. 2. Find a relation R such that y is a power of x if A = {1, 2, 3} and B = {1, 4, 5, 9}