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Main
UQU Programs
Mathematics

Foundation of Mathematics

MTH1201
4 hours English

Foundation of Mathematics

Foundation of Mathematics
1 Principle of Mathematical Logic: Statements, negation of statement, Connectives and truth tables, Methods of proof (direct proof, proof of the contrapositive, proof by contradiction, proof by counter-example, proof by mathematical induction)
2 Sets and algebra of sets: Method of defining a set (listing method, characteristic property), finite and infinite sets. Membership and inclusion, Universal and existential quantifiers. Power set, Algebra of sets (union, intersection, universal set, complement of a set, symmetric difference, De Morgan’s laws, Venn diagrams, repetition, sets of numbers (N, Z, Q, R and C)
3 Cartesian Product and Relations on Sets: Cartesian product of sets, ordered pairs, Binary relations on sets, reflexive, symmetric, transitive relations, Skew-symmetric (Anti-symmetric), Equivalence relation, ordered relation, Partition of sets and equivalence classes, Partial ordered relation, Inverse of relation, Composition of relations. Diagrams of relations
4 Mappings: Definition of mapping, Image of mapping, Inverse image of mapping Special types of mappings (injective (1-1), surjective (onto), bijective (1-1 and onto), Identity mapping, Composition of mappings, Bijection mappings as permutations, inverse of mapping. Equivalence of sets
5 Binary Operations: Definition and examples of binary operations, closure of a binary operation, commutative and associative operations, Identity element, Inverse of element, Systems of two operations, Homomorphism between two closed algebraic systems
6 Introduction to Groups: Semigroups (definition and examples), Groups (definition and examples), Subgroups (definition and examples), examples of groups and subgroups
7 Introduction to Rings and Fields: Definition and examples of rings. Definition and examples of fields, some properties of rings and fields. Ring of polynomials
8 Others: Revision and activities, Exam, quizzes
1.1 Mapped to: K1, K2

Explain concepts from elementary mathematical logic and compose truth tables of various propositional forms highlighting connectives, equivalence of propositions, contradictions, and tautologies.

Teaching Strategy Lectures
Assessment Methods Exams, quizzes
1.2 Mapped to: K1, K2

Distinguish between universal and existential quantified statements as well as their negations both in meaning and in use.

Teaching Strategy Lectures
Assessment Methods Exams, quizzes
1.3 Mapped to: K1, K2

Identify and use a suitable method of proof (direct proof, proof by contradiction, proof by contraposition, proof by cases, proofs involving quantifiers or their negations, and mathematical induction) in order to show the truth or falsehood of a given mathematical statement.

Teaching Strategy Lectures
Assessment Methods Exams, quizzes
2.1 Mapped to: S1, S2, S5, S6

Explain basic set-theoretic concepts (belonging, containment, equality, power set, indexed families of sets, and famous sets of numbers), and perform operations on sets (intersection, union, complement, difference, and symmetric difference).

Teaching Strategy Lectures
Assessment Methods Exams, quizzes
2.2 Mapped to: S1, S2, S6

Discuss properties of relations and resulting notions (equivalence classes, partitioning, and ordering).

Teaching Strategy Lectures
Assessment Methods Exams, quizzes
2.3 Mapped to: S1, S2, S5, S6

Explain properties of functions and related concepts such as injective, surjective, bijectivity, images and preimages of sets, inverse function equivalent sets, cardinality, finite and infinite sets.

Teaching Strategy Lectures
Assessment Methods Exams, quizzes
2.4 Mapped to: S1, S2, S5, S6

Illustrate properties of binary operations (associativity, commutativity, the identity, and invertibility), and give examples and counterexamples of relevant algebraic structures, and define homomorphism groups, rings, fields

Teaching Strategy Lectures
Assessment Methods Exams, quizzes
3.1 Mapped to: V1

Prepare for success in disciplines which rely on foundations of mathematics, which is the key to understand most of mathematical subjects.

Teaching Strategy Lecture/ Individual or group work
Assessment Methods Exams, quizzes
3.2 Mapped to: V2

Generalize mathematical concepts in problem-solving.

Teaching Strategy Lecture/ Individual or group work
Assessment Methods Exams, quizzes
Assessment
Weight
Midterm exam
30%
Homework and assignments
20%
Final Exam
50%
Total
100%