MATH 301

Math 301: Introduction to Mathematical Structures

Prerequisites: MATH 172

Credit Hours: (3)

An introduction to the foundations of modern mathematics via the study of the analytic structure of the real line. Topics include propositional logic, set theory, relations and functions, mathematical induction, the completeness axiom, and sequences and series. The methods of proof and problem solving for upper-division coursework are emphasized throughout the course.

Note(s): Students with prior credit for MATH 300 will not receive credit for MATH 301.

 

Content

1. Logic (propositions, logical connectives, truth tables, tautologies, and contradictions)
2. Methods of proof including direct proof, proof by contraposition, proof by contradiction, and proof by mathematical induction.
3. Set Theory (definitions and notations, set operations, difference and complement, Venn diagrams, Cartesian product, and partitions of sets)
4. Relations, equivalence relations, equivalence classes, and partial orders
5. Functions (image, range, domain, inverse image, surjective, injective, bijections, function composition, right inverse, left inverse, inverse functions, and permutations)
6. The Well-Ordering Principle
7. Completeness axiom (supremum and infimum)
8. Sequences (convergence, boundedness, monotonicity, subsequences, including Bolzano-Weierstrauss, and the Cauchy criterion)
9. Tests of convergence of numerical series (time permitting)

 

Detailed Description of Conduct of Course

This course may be offered in a variety of instructional formats, including a traditional lecture-based approach, a flipped classroom model, or other forms of Inquiry-Based Learning (IBL). Students may be asked to engage with readings, video lectures, or preparatory exercises. Some instructors may incorporate group work, encouraging students to work together in small teams to develop proofs, analyze examples, and present their reasoning.

 

Student Learning Outcomes

Students will be able to:

1. Define and apply the basic concepts of logic, include propositions, quantifiers, and logical reasoning.
2. Use the principles of set theory to describe and manipulate sets, subsets, unions, intersections, and complements.
3. Analyze and construct relations and functions, identifying key properties such as injectivity, surjectivity, and bijectivity.
4. Classify objects using equivalence relations and equivalence classes.
5. Explain the structure of the real number system, including order completeness, and density properties.
6. Apply the completeness property of the real line to solve problems and prove foundational theorems.

 

Assessment Measures

Student performance in this course will be evaluated through a combination of graded components that may include homework assignments, quizzes, exams, class participation, attendance, and projects and presentations.

 

Other Course Information

This course is intended as a requirement for math majors and an advanced elective for math minors and other related fields.

 

Review and Approval

November 20, 2025