MT4516 Finite Mathematics
有限数学代写 2. (a) Define what it means for two Latin squares to be orthogonal. [1] (b) Write down two orthogonal Latin squares of order 4. [3]
EXAM DURATION: 2 hours
EXAM INSTRUCTIONS: Attempt ALL questions.
The number in square brackets shows the maximum marks obtainable for that question or part-question.
Your answers should contain the full working required to justify your solutions.
PERMITTED MATERIALS: Non-programmable calculator
YOU MUST HAND IN THIS EXAM PAPER AT THE END OF THE EXAM PLEASE DO NOT TURN OVER THIS EXAM PAPER UNTIL YOU ARE INSTRUCTED TO DO SO.
2. 有限数学代写
(a) Define what it means for two Latin squares to be orthogonal. [1]
(b) Write down two orthogonal Latin squares of order 4. [3]
(c) State what it means to standardise a Latin square. [1]
(d) Let A and B be orthogonal Latin squares, and let A∗ and B∗ be the standardisations of A and B. Prove that A∗ and B∗ are orthogonal. (You do not need to prove that A∗ and B∗ are Latin squares). [3]
3. 有限数学代写
(a) Write down the definition of a projective plane. What is the difference beween an affine plane and a projective plane? [4]
(b) Let F be a finite field, and F3 a 3-dimensional vector space over F. Let P be the set of all 1-dimensional subspaces of F3 , and let L be the set of all 2-dimensional subspaces of F3 . We denote (P,L) by PP(F).
(i) Prove that PP(F) is a finite projective plane. [3] 有限数学代写
(ii) In PP(Z5), find the intersection of the lines
l1 = {(x, y, z) : x + 2y + 3z = 0}
and
l2 = {(x, y, z) : 2x + z = 0}.
[2]
(iii) In PP(Z5), find an equation defining the line containing the points {(t, 3t, 2t) : t ∈ Z5} and {(4t, 3t, t) : t ∈ Z5}. [2]
(iv) Determine the number of points and lines of PP(Zp). How many lines contain each point? Justify your answers. [4]
4. 有限数学代写
(a) State the definition of a (v, b, r, k, λ)-design. [2]
(b) Prove that the parameters of a (v, b, r, k, λ)-design must satisfy vr = bk. [2]
(c) There is a (7, 7, 3, 3, 1)-design whose blocks include {1, 2, 3}, {2, 5, 6} and {1, 6, 7}. Find its other blocks. [2]
(d) Does there exist a (25, 30, 6, 5, 1)-design? Does there exist a (9, 9, 3, 3, 2)- design?
You may use results from lectures without proof, provided that they are clearly stated. [4]
