 ## MT4516 Finite Mathematics

### 1.The incidence matrix A of an affiffiffine plane of order N = n 2 has rows indexed by the points, columns indexed by the lines, and entries A =

(aij )N×n2+n given by

aij = 1

if Pi lj

aij = 0    otherwise.

(i) Determine the value of AAT in terms of the N × N identity matrix

I and the N × N matrix J with all entries 1.   北美数学math作业代写

(ii) Show that (AT A)ij ∈ {0, 1, n} for all i and j, and determine the number of 0s, 1s and ns in each row of AT A.

2.(i) Write down the incidence matrix A of the fifinite affiffiffine plane of order 4 given in Figure 4.1 of the notes.

(ii) By calculating AAT and AT A directly, demonstrate that the results of Question 1 hold in this case.

### 3.Find an example of a (4, 4, 3, 3, 2)-design and an example of a (13, 13, 4, 4, 1)– design.  北美数学math作业代写

4.There is a (7, 7, 3, 3, 1)-design whose blocks include {1, 2, 4}, {2, 3, 5} and {3, 4, 7}. Find its other blocks.

5.Does there exist a (v, b, r, k, λ)-design with v = 14, b = 7 and r = 4?

### 6. Let X be a set of order v and let B be a set of k-element subsets of X (blocks).  北美数学math作业代写

Suppose that every pair of elements of X is contained in r blocks of B. Prove that, for each element of X, the number of blocks which contain the element is a constant.

7.Show that in a (v, b, r, k, λ)-design, if 2λ = r and b is divisible by 4, then λ is even.

8.Let (P,L) be a fifinite affiffiffine plane of order m2 and fifix t (2 t m). Let  X = P, and let B be the set of all sets of t collinear points. Prove that

(X, B) is a (v, b, r, k, λ)-design, and determine the values of the parame-ters v, b, r, k and λ.