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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.
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.
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.
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]
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
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
n+2 = a
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)
a) Let Fn denote the nth Fibonacci number. Show by induction that every natural number is
expressible as a sum
Fn1 +Fn2 +···+Fnk
where ni −ni+1 > 1 for each i ≥ 1.
b) Find the number of ways of tying up 7 different books into 4 bundles if the order of the books
in the bundles does not matter.
c) Give examples of trivially true and vacuously true statements.
a) Which of the following sentences are statements? Give reasons for your answer.
i) Are the buses running today?
ii) What a pleasant weather!
iii) x2 +1 = 0 for a real value of x.
iv) Every odd number is a prime.
b) Let t(n) denote the number of ways in which 2n tennis players can be paired to play n matches.
Determine a recurrence equation for the sequence t(n).
c) Show that any tree with exactly two vertices of degree 1 is a path.
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