# Mathematics I

Nelson Mandela University

## Course Description

• ### Course Name

Mathematics I

• ### Host University

Nelson Mandela University

• ### Location

Port Elizabeth, South Africa

Mathematics

• ### Language Level

Taught In English

• ### Course Level Recommendations

Lower

ISA offers course level recommendations in an effort to facilitate the determination of course levels by credential evaluators.We advice each institution to have their own credentials evaluator make the final decision regrading course levels.

### Hours & Credits

• Host University Units

15
• Recommended U.S. Semester Credits
1
• Recommended U.S. Quarter Units
1

## Syllabus

PURPOSE
To introduce learners to a selection of topics from Discrete Mathematics which are relevant to study and application in the field of Information Technology.

LEARNING OUTCOMES

• Define and identify the elements of logical expressions (atomic propositions, logical operations - negation, conjunction, disjunction, implication, equivalence, compound propositions).
• Demonstrate knowledge of logical variables, logical operators and operator precedence by formalizing natural language statements as syntactically and semantically correct logical expressions and by evaluating logical expressions.
• Produce truth tables for compound expressions.
• Define tautology, contradiction, contingency, logical equivalence, logical implication.
• Use truth tables to verify/refute the aforementioned.
• Use the Laws of Logic to verify/refute the aforementioned and simplify expressions.
• Use deductive proofs (Rules of Inference) to test the validity of arguments.
• Define sets, including the empty set, and the universal set with specific reference of the standard universal sets.
• Describe sets by enumeration and comprehension.
• Define and evaluate set cardinality.
• Define and use set equality and set inclusion.
• Define and obtain power sets and evaluate their cardinality.
• Define and use set operations.
• Know and apply the Principle of Inclusion and Exclusion.
• Recall and use the laws of set theory to simplify set algebraic expressions and to prove set identities.
• Explain what sequences are.
• Evaluate the general term of, and enumerate a sequence.
• Recognize and describe the difference between recursive and non-recursive definitions.
• Define sequences by means of recursive definitions and enumerate sequences defined recursively.
• Recall and use the division theorem.
• Define and establish divisibility.
• Recognize and use the ?mod? function.
• Define modular congruence and perform congruence arithmetic.
• Define prime and composite numbers, demonstrate use of the Sieve of Eratosthenes, and find prime factorisations using trial division. Knowledge and use of the smallest prime factor theorem is essential.
• Recall the Fundamental Theorem of Arithmetic.
• Define the greatest common denominator, determine by factorisation and by the Euclidean algorithm.
• Define the term coprime (relatively prime) and test for coprimality.
• Recall the linear combination theorem and express the gcd as a linear combination.
• Recall the theorem regarding the divisibility of congruences, define and obtain the inverse modulo m and define and solve linear congruences.
• Demonstrate the application of number theory to: hash functions, pseudorandom number generators, and simple ISBN check digit evaluation and the correction of single digit errors.
• Discuss and analyze cryptosystems with reference to: encryption and decryption, keys and keyspaces, shift ciphers, and affine ciphers.
• Define and recognize matrices and vectors and demonstrate knowledge of the notation used to refer to matrices, vectors, and their elements.
• Demonstrate knowledge of matrix terminology i.e. element, row vector, column vector, square matrix, null matrix, identity matrix, unit matrix, diagonal element, principal diagonal, diagonal matrix, symmetric matrix, triangular matrix.
• Evaluate matrix algebra expressions involving equality, transposition, addition, subtraction, scalar multiplication, vector product, matrix product, matrix power, including the join, meet, product and powers of Boolean matrices.
• Define, explain and recognize essential graph theoretical terminology.
• Know and apply selected graph theoretical theorems.
• Represent graphs using adjacency matrices and know and apply essential theorems relating to their use to answer questions regarding connectivity of graphs.
• Use Dijkstra's algorithm to solve shortest path problems.
• Define and identify trees, subtrees, and rooted trees and with regard to the vertices of rooted trees: root, parent, child, sibling, ancestor, descendant, internal vertex, leaf, level, height, balanced tree, m-ary tree, full m-ary tree, and binary tree.
• Recall and use the theorems relating to trees.
• Perform the three tree traversal algorithms: inorder, preorder, postorder.
• Demonstrate the use of binary trees for binary search trees, prefix codes and expression trees.
• Define spanning trees and minimum spanning trees and produce minimum spanning trees by means of Prim?s algorithm.

CORE CONTENT

• Propositional logic.
• Set theory.
• Sequences and recursion.
• Number theory and applications.
• Elementary matrix algebra.
• Graph structures and applications.
• Tree structures and applications.