MATH 100

BS Mathematics

1

6

24

36

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## Content

#### Objective

1. Write a mathematical text clearly and to present it with a logical sequence 2. Use the logical and mathematical symbols correctly. 3. Make direct proofs and by the absurdity as well as proofs by induction. 4. Give counterexamples in logic. 5. Correctly use the language of the set theory. 6. Control the operations on the sets. 7. Define the mappings and their various types and give examples. 8. Find the composite of mappings and the inverse of a bijective mapping. 9. Iidentify an equivalence relation and an ordered relation and to give examples 10. Define lub and glb of a non empty subset in an ordered set. 11. Define a group, a ring and a field and give examples.

#### Content

Elementary Logic: assertions, logical connectives, logic in Proofs, methods of proof, quantifiers. Sets: definitions, set operations, properties, cartesian product, family and index set, intersection and union of a family of subsets of a set. Mappings: basic properties, composite of mappings and, types of mappings . Relations: equivalence relations, partitions, ordered relations, special orderings, lub and glb. Finite Set: definition, cardinal of a finite set, basic properties. Groups: binary operations, groupoids, groups, subgroups, homomorphisms. Rings and Fields: definition, properties, subrings, ideals, Fields. Comment: It is necessary to avoid teaching a hard course in logic. Logic that the students are invited to use is that dictated and expressed in a clear way with the ordinary language. It is based mainly on the principle of contradiction. The students should be invited to use the least possible the logical symbols and to give them the example by avoiding oneself doing it. one will insist specially on the rules of negation of a logical sentence and on the fact that to write equivalence P  Q it is necessary to be sure that the implications P  Q and Q P are both true at the same time. One will also insist on correct use of quantifiers. One will clarify the various methods of proofs. One will multiply the exercises with an aim of leading the students to control these concepts and these practices. Concerning the sets, one will avoid any axiomatic construction of their theory. One will insist on correct use of the notions and the language of the set theory without entering many details. Concerning the mappings, one will insist on the concept of surjectivity and injectivity and multiply the exercises on this concept. As for the equivalence relation, one will insist on the relation between the equivalence relations and the partitions without studying the partitions in detail. One will show to the students, using examples, that each equivalence relation R on a set S, the union of distinct R-equivalence classes is S. Concerning the ordered relations, one will multiply the examples. It will be noted that the relation a divides b is a relation of order on ℕ but it is not on ℤ. One will multiply the exercises on ordered sets specially in ℕ, ℤ and IR. Concerning the algebraic structures, one will satisfy to define the structure of a group, the concept of a sub-group, the structure of a ring and that of a field and one will multiply the examples of these structures.

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