CAS-025 Lecture Notes - Discrete Mathematics, Graph Theory, Pigeonhole Principle

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labeeb246

22 Feb 2023

School

American University

Department

College Of Arts & Sciences

Course

CAS-025

Professor

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Document Summary

Discrete mathematics is the study of mathematical structures that are fundamentally discrete and not continuous. It is an essential foundation for computer science, mathematics, and engineering. Understanding discrete mathematics is crucial for aspiring computer scientists and mathematicians, as well as anyone who wants to build a solid foundation in mathematics. In this comprehensive guide, we will explore basic discrete mathematics and its applications. Discrete mathematics is a branch of mathematics that deals with mathematical structures that are discrete, as opposed to continuous. This means that they have distinct values and are not continuous, like real numbers. The structures studied in discrete mathematics include sets, relations, functions, graphs, logic, and combinatorics. The applications of discrete mathematics are vast and diverse, ranging from computer science to cryptography and biology. In discrete mathematics, sets are collections of distinct objects. They can be finite or infinite, and their elements can be anything.

Related Questions

Recursion Question 5. In the following question, you will be asked to program the following using recursion. If you have an iterative solution, you may use it, but the most you can get for the question will be half credit. (5) points) Consider the following mathematical function h which operates on integers. (1, if n < 2 h(n) = n(n-1) +n if n > 2 and n is even han -2) +2*n if n > 2 and n is odd In Java, use recursion to write a method h which takes as input an int n and returns h (n). public static int h(int n) {

Do Enzymes Physically Attach to Their Substrates?

When scientists first began to examine the chemical activities of organisms, no one knew that biochemical reactions were catalyzed by enzymes. The first enzyme was discovered in 1833 by French chemist Anselme Payen. He was studying how beer is made from barley: First, barley is pressed and gently heated so its starches break down into simple two-sugar units; then yeasts convert these units into ethanol. Payen found that the initial breakdown requires a chemical factor that is not alive and that does not seem to be used up during the process-a catalyst. He called this first enzyme diastase (we call it amylase today).

Did this catalyst operate at a distance, increasing the reaction rate all around it, much as raising the temperature of nearby molecules might do? Or did it use physical contact, actually attaching to the molecules whose reaction it catalyzed (its "substrate")?

The answer was discovered in 1903 by French chemist Victor Henri. He saw that the hypothesis that an enzyme physically binds to its substrate makes a clear and testable prediction: In a solution of substrate and enzyme, there must be a maximum reaction rate. When all the enzyme molecules are working full tilt, the reaction simply cannot go any faster, no matter how much more substrate you add to the solution. To test this prediction. Henri carried out the experiment whose results you see in the graph, measuring the reaction rate (V) of diastase at different substrate concentrations (S).

Analysis

1. Making Inferences As S increases, does V increase? If so, in what manner-steadily, or by smaller and smaller amounts? Is there a maximum reaction rate?

2. Drawing Conclusions Does this result provide support for the hypothesis that an enzyme binds physically to its substrate? Explain. If the hypothesis were incorrect, what would you expect the graph to look like?

3. Further Analysis If the smaller amounts by which V increases are strictly the result of fewer unoccupied enzymes being available at higher values of S, then the curve in Henri's experiment should show a pure exponential decline in V-mathematically, meaning a reciprocal plot (1/V versus 1/S) should be a straight line. If some other factor is also at work that reacts differently to substrate concentration, then the reciprocal plot would curve upward or downward. Fill in the reciprocal values in the table to the right, and then plot the values on the lower graph (1/S on the x axis and 1/V on the y axis). Is a reciprocal plot of Henri's data a straight line?