Lecture 1 full notes

Introduction to Mathematics

• Sequences and Recursion: Focus on understanding sequences through recursive definitions.

• Why Study Mathematics:

  • Considered the language of Science, Technology, Engineering, and Mathematics (STEM).

  • Mastery of mathematical language improves fluency in STEM fields.

  • Learning involves memorizing vocabulary, simple phrases, and grammar structures; similarly, math requires practice and familiarity.

• What is Discrete Mathematics?:

  • Studies discrete sets such as integers, as opposed to continuous sets like real numbers (R).

  • Example: Continuous problem - Finding maximum value on an interval.

Examples of Discrete Problems

• Counting and Probabilities:

  • How many student identifiers have non-repeating digits?

  • Estimating time to guess a Bitcoin private key.

• Travelling Salesman Problem:

  • Finding the shortest route to visit multiple cities.

• Relevance in Computer Science:

  • Most IT and CS problems fall into discrete mathematics, with some exceptions like image transformations.

• Logic in Mathematics:

  • Upcoming topics include converting arguments into symbolic forms and using Boolean algebra, particularly useful in programming.

Sequences

• Definition: Arranged set of terms in a definite order.

  • Examples:

    • Sequence of integers: 1, 2, 3, 5, 8.

    • Sequence of real numbers: 3, 3.1, 3.14, 3.141, 3.1415.

  • Functions: sin(x), sin²(x), sin(3x).

• Terms in a Sequence:

  • Denoted as t_n, where n indicates the term's position.

Recursive Definitions of Sequences

• Examples of Sequences:

  • Even numbers: 2, 4, 6, 8, denoted as t_n = 2n.

  • Odd numbers: 1, 3, 5, 7, with recursive definition:

    • t_1 = 1

    • t_n = t_(n-1) + 2.

• Importance of Recursive Definitions:

  • Useful when the direct pattern is not evident or when direct formulas are complicated.

Recursive Definition of Odd Numbers

• Base Case:

  • t_1 = 1.

• Recurrence Relation:

  • t_n = t_(n-1) + 2.

• Challenges of Direct Formulas:

  • Complex formulas (e.g., Fibonacci sequence) may be difficult to prove.

Arithmetic Sequences

• Definition: Sequence where the difference between successive terms is constant.

• Example:

  • Arithmetic sequence: t_n = a + (n-1)d, where d is the common difference.

• Recursive Definition:

  • t_n = t_(n-1) + d.

Geometric Sequences

• Definition: Sequence where each term is a constant multiple (common ratio) of the previous term.

• Example: t_n = a * r^(n-1), where r is the common ratio.

• Recursive Definition:

  • t_n = t_(n-1) * r.

Exercises

• Instructions to find first five terms for given sequences.

  • For example, finding terms for t_n = 4h + 1 leads to 4, 5, 7, 11, 15.

  • Another task: given t = 3 + t_(n-1) for n > 2, find terms recursively.

Exercise Solutions

• First Five Terms from Formula:

  • For t_n = 4h + 1, solutions are 4, 5, 7, 11, 15.

  • For recursive definition: if t_n = 3 + t_(n-1), with t_1 = 2, then terms yield similar patterns.