The unit 18 discrete mathematics in the computing for HND diploma emphasises the concept of foundation mathematics. They cover the concepts which are crucial for problem-solving and software. This unit begins with a set of functions and theories which the students can acquire to improve their algebra operations performance. In addition, the students can apply the De Morgan law principles and the calculation of cardinalities. These skills are significant for recognising the structure of relational data, algorithm design, and database operations. For example, the solving of problems includes the complement of sets, intersections and unions in questions with optimisation of the retrieval of data. Multiple functions of mathematics, such as domain reigns identification and invertibility, are identified with the help of scenarios in the real world, like system mapping and encryption.
Another area of this unit is to cover Boolean algebra and craft theory. The theory of graphs identifies structures such as shortest path algorithms, trees of binary numbers, for example, algorithm of Dijkstra and the cycle of networking. The main application of Boolean algebra is in logic circuits, which are used to simplify the expressions and generate the digital system with Truth tables. Its students will identify the Hamiltonian and Eulerian cycle for the identification of roots in logistics and networking. For example, the scenarios of the truth model of tables, such as the rainfall system of detection and seat belt warnings, directly provide a connection to the conditional programming and hardware design. Circuit simplification and K-map techniques further improve the efficiency in the task of electronic engineering.
Unit 18 also deals with abstract algebra, which includes fields, rings and groups for solving the problems of complex computation. The concepts include cryptography, modular arithmetic and error-correcting codes of error in cybersecurity. Multiple techniques of proof, such as distributive law induction, algorithm validation, will strengthen logical reasoning. With the help of integration in discrete structures, the student will increase their tools for the optimisation of databases, development of efficient software and secure networks. All of these tactics make the unit significant for a career in cybersecurity, system engineering and data sciences.
Aims of the unit
Here are the main objectives of unit 18, discrete maths in computing diploma of HND, through which you can get the idea.
- To apply and understand the set of functions and theories which are relevant to the software, it enables them to perform the operations of the algebraic set and identify the relationships of mathematics.
- To identify the structures of mathematics utilising the theories of graphs that include the problems of modelling by applying algorithms and binary trees. It includes the solution of the shortest path with Dijkstra.
- To solve and investigate the practical issues with the help of Boolean algebra, which includes simplifying circuits with logic, constructing and creating digital systems with truth tables.
- To apply and explore concepts from algebra abstracts, such as fields, rings and groups, to support cryptography and computing with advancement.
- To generate techniques of proof that include induction of mathematics for property validation, such as the distributive law and De Morgan`s law in the theory set.
- To determine and assess the presence of Hamiltonian and Eulerian cycles in graphs to improve the logistics and networking with problem-solving.
- To give strength to the logical reasoning and skills of analytics for utilising databases, development of efficient software solutions and secure designing of networks.
Learning Outcomes
Unit 18: discrete maths contains some main learning outcomes of the HND course for learners of computing are demonstrated below:
LO1: Examine set theory and functions applicable to software engineering.
- Set of theory:
- Set operations and sets.
- Set theory algebra
- Identities of the set and identities of the proof
- Manipulative functions of bags
- Functions:
- Mapping, range and domain
- Inverse function and inverse relations
- Subjective and injective functions and relations of transitive.
LO2: Analyse mathematical structures of objects using graph theory.
- Theory of graphs:
- Characterisation and structure of the graph
- Rooted trees and spanning trees
- Hamiltonian and Eulerian graphs
- Edge and vertex graph colouring
- Directed graph:
- Undirected and directed graphs
- Trails, paths and walks with the shortest paths
LO3: Investigate solutions to problem situations using the application of Boolean algebra.
- Boolean algebra stages of binary system, for example, open or close, high or low, off or on.
- Problem identification and outputs and inputs of labelling. Truth table production corresponding to the situation with the problem.
- Equations:
- Identification of the truth table according to the equation of Boolean
- Simplification of Boolean equations utilising the methods of algebra. Boolean equation representation utilising the gates of logic.
LO4: Explore applicable concepts within abstract algebra.
- Structures of algebra
- Associated properties and binary operations
- Associative operations and commutative operations
- Structures and algebraic structures
- Groups
- Semigroups and monoids, group codes and group families, groups introduction
- Morphisms and substructures
Assessment Criteria
The assessment criteria of unit 18: discrete maths are associated with the learning outcomes according to the HND diploma in computing.
LO1: Examine set theory and functions applicable to software engineering
- AC 1.1 Perform algebraic set operations in a formulated mathematical problem.
- AC 1.2 Determine the cardinality of a given bag (multiset).
- AC 1.3 Determine the inverse of a function using appropriate mathematical techniques.
- AC 1.4 Formulate corresponding proof principles to prove properties about defined sets.
LO2: Analyse mathematical structures of objects using graph theory.
- AC 2.1 Model contextualised problems using trees, both quantitatively and qualitatively.
- AC 2.2 Assess whether an Eulerian and Hamiltonian circuit exists in an undirected graph.
- AC 2.3 Construct a proof of the Five Colour Theorem.
- AC 2.4 Use Dijkstra’s algorithm to find a shortest path spanning tree in a graph.
LO3: Investigate solutions to problem situations using the application of Boolean algebra.
- AC 3.1 Diagram a binary problem in the application of Boolean algebra.
- AC 3.2 Simplify a Boolean equation using algebraic methods.
- AC 3.3 Design a complex system using logic gates.
- AC 3.4 Produce a truth table and its corresponding Boolean equation from an applicable scenario.
LO4: Explore applicable concepts within abstract algebra.
- AC 4.1 Describe the distinguishing characteristics of different binary operations that are performed on the same set.
- AC 4.2 Validate whether a given set with a binary operation is indeed a group.
- AC 4.3 Explore, with the aid of a prepared presentation, the application of group theory relevant to your given example.
- AC 4.4 Determine the order of a group and the order of a subgroup in given examples.
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