Discrete Math: Matrix Operations (proofs)

Prove or disprove that, for matrices A,B,C for which the following operations are defined: a. A*(B+C) = A*B + A*C b. A+(B*C) = (A+B)(A+C)

Determine reflexive, symmetric, antisymmetric, transitive, partial order and equivalence.

Let R = {(1,1)(3,1)(2,2)(1,2)(3,3)(3,2)} on Z = {1,2,3} Is R reflexive? Why? Is R Symmetric? Why? Is R antisymmetric? Why? Is R transitive? Why? Is R a partial order? Why? Is R an equivalence relation?

Discrete Structures

1. Consider the sequence of triangles Ti, i >= 2: T2 is simply a triangle sitting upright, on its base. T3 is T2, except that an additional straight line is drawn from the upper vertex, down to somewhere on the base. For each Ti+1, one more line is added to triangle Ti (such that each line meets the base at a different point). For each Ti we define ai to be the number of triangles ?contained? in Ti, each comprising 1 or more areas of Ti. So, e.g., a2 = 1, but a3 = 3: the triangle on the left, the triangle on the right, and the whole triangle. a. Find a recurrence relation for the sequence ai. b. Find a close-form solution for the sequence ai. (It may be easier to do this part first... click for more

Discrete structures

6. A string that contains only 0s 1s and 2s is called a ternary string a) find a recurrence relation for the number of ternary strings that contain two consecutive 0s b) what are the initial conditions c) how many ternary strings of length six contain two consecutive 0s The next 3 problems deal with a variation of the Josephus problem described by Graham, Knuth, and Patashnik in [GrKnPa94].? This problem is based on an account by the historian Flauvius Josephus, who was part of a band of 41 Jewish rebels trapped in a cave by the Romans during the Jewish-Roman war of the first century.? The rebels pre-ferred suicide to capture; they decided to form a circle and to repeatedly count o... click for more

Discrete Structures

11) Use generating functions to determine the number of different ways 12 identical action figures can be given to five children so that each child receives at most three action figures. 12) Use generating functions to find the number of ways to select 10 balls from an urn containing red, white and blue balls if: a. The selection has at least two balls of each color. b. The selection has at most two balls of each color. c. The selection has an even number of red balls. 13) Determine whether the relation R on the set of all real numbers is reflexive, symmetric, antisymmetric, and/or transitive, where (x,y) E R, if and only if: a) x + y = 0 b) xy = 0 14) Find: a) R1 U... click for more