Application of Discrete Mathematics in Engineering
- 1. PPT Topic – Application of Discrete Mathematics in Engineering
- 2. Discrete Mathematics for Engineering Discrete: 2 a : consisting of distinct or unconnected elements : NONCONTINUOUS b : taking on or having a finite or countably infinite number of values <discrete probabilities> <a discrete random variable> — Meriam Webster Online. Mathematics: Science of structure, order, and relation that has evolved from counting, measuring, and describing the shapes of objects. —Encyclopaedia Britanica Online. for: 1 a : used as a function word to indicate purpose —Meriam Webster Online. Engineering: 2 a : the application of science and mathematics by which the properties of matter and the sources of energy in nature are made useful to people b : the design and manufacture of complex products <software engineering> —Meriam Webster Online.
- 3. Introduction What is discrete math? How can it help us? Propositional Logic Applications: Digital logic, if-statements in computer programs Sets and Boolean Algebra Set building notations. Set operations. Reasoning about sets. Application: Hamming codes Predicate Logic Application: preconditions, postconditions, assertions; temporal properties of systems. Integers, Mathematical Reasoning, and Induction Application: RSA encryption. Counting and Recurence Relations Application: Counting AVL trees. Functions and Relations Application: Cryptology. Relational databases. System models. Functional models of computer programs. Relational specifications of computer programs. Graphs and Trees Applications: Shortest distance/maximal reliability/etc. PCB testing. Maximum flow. Minimal spanning trees for power or data networks. Huffman encoding. Models of Computation Application: Design of discrete systems. Design of languages. Safety analysis of systems.
- 4. What is discrete math? • The real numbers are continuous in the senses that: between any two real ∗ numbers there is a real number. • The integers do not share this property. In this sense the integers are unconnected or “discrete” Discrete math is the study of mathematical objects that are discrete in this sense. Some discrete mathematical concepts: • Integers: Between two integers there is not another integer. • Propositions: Either true or false, there are no 1/2 truths (in math) • Sets: An item is either in a set or not in a set, never partly in and partly out. • Relations: A pair of items are related or not. • Networks (graphs): Between two terminals of a direct network connection there are no other terminals.
- 5. Relevence to ECE Discrete math is relevent to computer and electrical engineering because we often deal with objects with discrete properties. Consider a digital video clip: • Pixels are discrete objects in space. • Frames are discrete in time • Colour frequency is discretized into 3 hues: red, green, and blue. • For each hue, intensity comes in discrete levels, e.g. 256 per colour. In this case the discrete values represent continuous time, space, frequency, and intensity variables. Often discrete variables are used to represent discrete rather than continuous phenomena: • The states of a computer program are discrete. • The states of a digital hardware design are discrete. • Discrete structures (sets, functions, relations, trees, graphs) are very useful for representing data in computers. • Connecting a generator or load to a power grid is a discrete change. • Two computers in a network either are or are not directly connected.