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Discrete Optimization

Discrete Optimization is a branch of optimization which embodies a significant area of combinatorics that deals with discrete values, such as integers. There are two main branches of Discrete Optimization:

1. Combinatorial Optimization, which refers to problems which deal with combinatorial structures such as graphs and,

2. Integer Programming, which refers to problems where mathematical optimization only deals with integers.

However, although these subjects may be considered different branches of Discrete Optimization, they are in fact not completely isolated from each other as problems under Combinatorial Optimization can fall under Integer Programming and vice versa. However, with the growth in this area of Mathematics, both branches are often used in conjunction to optimize, in other words, to find efficient methods of constructing good solutions as well as measuring the quality of these particular solutions.

The applicability of this method extends into almost every facet of society, from scheduling planes to coordinating the production of steel to designing pharmaceutical drugs. The methodology itself of finding good solutions to these everyday problems includes a variety of mathematical techniques such as constructing tree-growing procedures, integer lattices as well as the analysis of algorithms. Thus, understanding Discrete Optimization is crucial for the study of Discrete Math as well as other disciplines which examines optimization.

Enumerations, Combinations, and Permutations

Enumeration Example Suppose ABC University has 3 different math courses, 4 different business courses, and 2 different sociology courses. Tell me the number of ways a student can choose one of EACH kind of course. Then tell me the number of ways a student can choose JUST one of the course. Enumeration - Handshakes Consider

Mathematic Custom Help

18. Of 100 clock radios with digital tuners and/or CD players sold recently in a department store, 70 had digital tuners and 90 had CD players. How many radios had both digital tuners and CD players? (Is the answer 70?) If not please give the answer and explain. 22. Of 50 employees of a store located in downtown Boston,

Auto Insurance Percentage of Coverage

Please help with the following problem. An auto insurance company classifies its customers in three categories: poor, satisfactory, and preferred. Each year, 30% of those in the poor category are moved to satisfactory and 5% of those in the satisfactory category are moved to preferred. Also, 5% of those in the satisfactory c

Smallest positive value

Suppose n = (2^5)(3^7)(5^9)(7^4) m where n and m are integers. What is the smallest positive value of m that makes n a perfect cube (multiply the number out, please).

Discussion Regarding Sampling

Question: How long does it take to travel to work Monday through until Friday for 10 days? Monday = 30 + 35 /2 = 65/2 = 32.5 or 32 ½ or 0.10485 Tuesday = 35 + 25 /2 = 60/2 = 30 or 0.09675 Wednesday = 30 + 30 = 60/2 = 30 or 0.09675 Thursday = 25 + 30 /2 = 55/2 = 27.5 or 27 ½ or 0.0887 Friday = 35 + 35 = 70 /2 = 35 or

Truth Tables, Implications, Contrapositives and Converses

(a) Use truth tables to prove that an implication is always equivalent to its contrapositive. Site an example where this is so. (b) Use truth tables to prove that an implication may not be equivalent to its converse. Site an example where this is so.

Relations : Reflexive, Symmetric and/or Transitive

Determine if the relation R on the set of all people is reflexive, symmetric and/or transitive where (x,y) "E" R if and only if x and y live within one mile of each other. NOTE: I cannot correctly indicate the symbol to show "is a member of" so I have used "E" in it's place.


If f o g are one-to-one, does it follow that g is one-to-one? Justify your answer.

Can someone give me a solution

3. Two cards are drawn form a standard deck of 52 cards. Explain why each of the following sets is not a Sample Space for this experiment. For each set, state all of the characteristics of a Sample Space which that set fail to have, giving an example of how it violates that necessary property of a Sample Space. (a) {at least 1


For any set B, B u B'=U True or False?

Minimizing Cost

There are two companies, the IL Company and the MO Company. They are trying to build a 3000 ft. tunnel connecting St. Louis, IL and St. Louis, MO. Each company will begin digging on their own side and eventually meet up with the other. The cost of digging x feet of tunnel for the IL Company is 1.95x^(2)+20x and it is 2.05x^(2)-3

12 students 3 ways

12 students are split 5 in room A, 4 in room B, and 3 in room C. How many different ways can this happen.