Ordered Pairs

Describe R by listing the ordered pairs in R and draw the digraph of this relation. See attached file for full problem description.

A binary relation R is defined in terms of a given matrix. Define what it means for a relation to be (a) reflexive, (b), antisymmetric, and (c) transitive. Also, determine which of these are properties of R.

For the set A = {a, b, c}, let R be the relation on A which is defined by the following 3 by 3 matrix M_R: ---------------------------------------- Row 1: 1 0 1 Row 2: 1 1 0 Row 3: 0 1 1 ----------------------------------------- Which of the properties (reflexive, antisymmetric, transitive) are satisfied by R? ...continues

A binary relation R is defined in terms of a given matrix. Determine the transitive closure of R.

For the set A = {a, b, c}, let R be the relation on A which is defined by the following 3 by 3 matrix M_R: ---------------------------------------- Row 1: 1 0 1 Row 2: 1 1 0 Row 3: 0 1 1 ----------------------------------------- Determine the transitive closure of R.

Let R be the relation defined on Matrix A. We have to draw the digraph of the transitive closure of R and use the digraph to explain connectivity. For complete description of the problem, please see the attached problem file.

Draw the digraph of the transitive closure of R and use the digraph to explain the idea of connectivity. Is this graph connected? What does connectivity mean? See attached file for full problem description.

Trees

1.Use the depth-first search numbering obtained in the indicated exercise to list the back edges in the graph. Use the file (5.3jpg) 2. Use Prim's algorithm to find a minimal spanning tree for each weighted graph. (Start at A) Give the weight of the minimal spanning tree found Use 5.2prims.jpg

A binary relation R is defined in terms of a given matrix. Determine whether R is a partial order. If it is, draw its Hasse diagram.

For the set A = {a, b, c}, let R be the relation on A which is defined by the following 3 by 3 matrix M_R: ---------------------------------------- Row 1: 1 0 1 Row 2: 1 1 0 Row 3: 0 1 1 ----------------------------------------- Determine whether R is a partial order. If it is, draw its Hasse diagram.

This problem introduces basic axioms of probability. The idea of intersection and union are examined. Simple examples are used to illustrate these ideas.

Q: Let A and B be two events defined on a sample space S such that P(A)=0.5 P(B)=0.25 P(A or B)=0.7 [P(A union B)=0.7] Find the following (1) P(A and B) [P(A intersect B)] (2) P(A^C and B) [P(A complement intersect B)] See word document for a cleaner version of the problem.

Product of Disjoint Cycles

In S(5) let pi=(245)(1354)(125). Write pi as a product of disjoint cycles and then answer the following questions. (a) Determine pi^2, pi^5, pi^(-1). (b) What is the order of pi? Why?

Proof methods & strategy

Problem: Prove that there is a positive integer that equals the sum of the positive integers not exceeding it. Is your proof constructive or nonconstructive?

Evaluating Functions

Please see the attached file for the fully formatted problems. Let S = {-1, 0, 2, 4, 7}. Find if i) ii) iii) iv)