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Discrete mathematics is defined as a branch of mathematics which deals with discrete mathematical structures. It does not support continuous mathematical structures and deals with various topics like discrete calculus, discrete analysis, operations research, combinatorics, probability, and many others. Few applications of the Discrete Mathematics are as follows:
- Data Structures
- Algorithm design
- Complexity theory
- Compiler design
- Relational database theory
- Mathematical logic
Discrete mathematics tells about the proof techniques and mathematical reasoning. It is extended through algorithms of computer programming to analyze the difficult infrastructure such as utility distribution, highway and traffic pattern, and many more. It interrelates with the other major disciplines such as physics, science, computer science, topology, algebra, etc. It is real world mathematics and essential to the college–level mathematics. Theory of computation and graph theory are considered as the parts of the discrete mathematics.
Discrete mathematics becomes popular in the today’s world due to its applications in CS. It is also known as the mathematics of computing as its concepts plays an important role in determining the problems in the section of computer science, such as cryptography, programming languages, computer algorithm, etc. Some of the core topics that involved in the Discrete Mathematics are listed below:
- Sets: it refers to the disordered collection of various elements. There are various types of set, including Subset, Proper subset, Finite set, Universal set, Overlapping set, Disjoint set, and many others.
- Functions: functions are allotted to the each element of the set.
- Predicate logic: it deals with predicates which contain variables.
- Counting theory: it involves the combination rule, counting rule and permutation rule.
- Probability: it refers to the sub-disciplines of mathematics that mainly concerned with the events that occur in the countable sample spaces.
- Rules of Inference: Rules of inference is used in acquiring the new statements from the given statement whose truth we already know.
- Propositional logic: it refers to a statement which is either true or false.
- Relation: it occurs between the objects of sets.
Extremal graph theory in discrete computing refers to the branch of graph theory that tells about the intrinsic structures of graphs which satisfy a definite property under the suitable conditions.
However, Extremal finite set theory is known as the quickly developing areas in Combinatorics. It has the several applications in the branches of CS and Mathematics such as Probability theory, Functional Analysis, Discrete Geometry, etc. It also involves the applications of probabilistic arguments in itself.
Here is an exposure to core Advanced topics in Discrete Mathematics:
- Upper-level set theory: set theory mainly concerned with the sets which are the collections of objects. Countable sets play an important role in the Discrete Mathematics.Major topics that involved in the Upper level set theory are Zermelo-Frankel Axioms, Topoi and Axiomatic set theory.
- Upper-level number theory: it concerned with the properties of numbers, mainly with the positive integers. It has applications in various areas such as Diophanyine equations, cryptanalysis, cryptography and many more. Major concepts that involved in this are Analytic Number theory, polynomials, number representations, arithmetic functions, finite fields and modular arithmetic.
- Upper-level logic: major concepts that involved in the Upper-level logic are second order logic and Godel’s incompleteness theorem.
Asymptotic Notation is known as the algorithm’s growth rate which enables user to analyze the algorithm’s running time by recognizing its behavior as the input size for algorithm increases.
Furthermore, Automata theory is a theory in discrete mathematics which concerned with the automata and abstract machines for solving the computational problems. Four major families of automata theory are Pushdown automata, Turing machine, Linear-bounded automata and Finite-state machine.
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- Theory of mathematical proof, specification of deductive proofs in an axiomatic system, types of proofs, generalization , falsification, inductive proof , Theory of sets, algebra of sets, functions, relations, counting, combinatorics , Matrices, inverse matrix, system of linear equations, Gauss elimination method, determinants
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