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In a lottery, players win a large prize when they pick four digits that match, in the correct order, four digits selected by a random mechanical process. A smaller prize is won if only three digits are matched. What is the probability that a player wins the large prize? What is the probability that a player wins the small prize?

determine that wheter the functions from real numbers to real numbers are one to one

f(n)=n^3

f(n)=n^2+1

f(n)=n^3

f(n)=n^2+1

Determine if the statements are valid arguments.

I just got my ticket to the concert but if the number of tickets sold is more than 75% of its capacity, I will not want to go because it will be too crowded and I don’t like it. If my friend Alex goes then I will go too. I can’t believe this but yesterday I heard that Alex got tickets too and is going. The concert is sold out as of this morning. After so much reflection, I will go to the concert.

I just got my ticket to the concert but if the number of tickets sold is more than 75% of its capacity, I will not want to go because it will be too crowded and I don’t like it. If my friend Alex goes then I will go too. I can’t believe this but yesterday I heard that Alex got tickets too and is going. The concert is sold out as of this morning. After so much reflection, I will go to the concert.

Mathematical Induction. Suppose that you know that a cyclist rides the first kilometre in an infinitely long road, and that if this cyclist rides one kilometre, then she continues and rides the next kilometre. Prove that this cyclist will ride every kilometre in that infinite road.

Explain 1. Game Tree 2. Decision Tree

Find the number of equivalence relations that can be defined on a set of 6 elements.

a) Obtain the Ferrar graph of the partition 8+7+6+5+5+3+2+1. Find the conjugate

partition. Is the partition self conjugate?

b) For the graph in Fig. Fig. 2 on the following page find a minimal colouring.

c) Is Petersen graph bipartite? Give reasons for your answer.

partition. Is the partition self conjugate?

b) For the graph in Fig. Fig. 2 on the following page find a minimal colouring.

c) Is Petersen graph bipartite? Give reasons for your answer.

A) Find the generating function of the recurrence

an = 4an−1 −4an−2 +1

with initial conditions a0 = 1, a1 = 1.

B) Express 3x4 +2x3 −2x2 +x in terms of [x]4, [x]3, [x]2 and [x]

an = 4an−1 −4an−2 +1

with initial conditions a0 = 1, a1 = 1.

B) Express 3x4 +2x3 −2x2 +x in terms of [x]4, [x]3, [x]2 and [x]

a) Let A be an 8×8 Boolean matrix (i.e. every entry is 0 or 1). If the sum of the entries in A is

51, prove that there is a row i and a column j in A such that the entries in row i and in column j

add up to more than 13. Further, show that there are at least 4 such pairs of rows and

columns.

b) If a planar graph has the degree sequence {2,2,3,3,4,4,4}, how many faces will it have? Draw

a planar graph with this degree sequence and number the faces to check your answer.

c) Give the order and the degree of the recurrence

a

2

n+2 = a

2

n +2an +4

Is the recurrence homogeneous?

51, prove that there is a row i and a column j in A such that the entries in row i and in column j

add up to more than 13. Further, show that there are at least 4 such pairs of rows and

columns.

b) If a planar graph has the degree sequence {2,2,3,3,4,4,4}, how many faces will it have? Draw

a planar graph with this degree sequence and number the faces to check your answer.

c) Give the order and the degree of the recurrence

a

2

n+2 = a

2

n +2an +4

Is the recurrence homogeneous?

a) Make a table of the values of the Boolean function

f(x1,x2,x3) = x2 ⊕(x1 ∧x3)

Write the function in DNF using the table.

b) Find the general form of the solution to a linear homogeneous recurrence with constant

coefficients for which the characteristic roots are 1 with multiplicity 1, −2 with multiplicity 2

and 2 with multiplicity 3. The relation also has a non-homogeneous part which is a linear

combination of n2n

and (−2)

n

f(x1,x2,x3) = x2 ⊕(x1 ∧x3)

Write the function in DNF using the table.

b) Find the general form of the solution to a linear homogeneous recurrence with constant

coefficients for which the characteristic roots are 1 with multiplicity 1, −2 with multiplicity 2

and 2 with multiplicity 3. The relation also has a non-homogeneous part which is a linear

combination of n2n

and (−2)

n