Join Peggy Fisher for an in-depth discussion in this video Prove with mathematical induction, part of Programming Foundations: Discrete Mathematics.
- [Voiceover] One of the most important types…of proof in discrete mathematics…is called mathematical induction.…This process allows you to verify a given theorem.…Induction is the proof technique…that is especially useful for proving statements…about elements in a sequence.…The two components of the inductive proof are…first, identifying the base case,…which establishes that the theorem is true…for the first value in the sequence.…Next, you identify the inductive step,…which establishes that if the theorem is true for k,…then the theorem holds true for k plus one.…
Therefore, if the base case is true…and the inductive step is true,…then the theorem holds for all natural numbers.…Let's take a look at proving an identity by induction.…I think mathematical induction…is probably best learned by doing.…So let's walk through an example.…Here, I have a theorem that says…for all values, n, greater than or equal to one.…In other words, all positive integers.…The sum of all the integers from one to n…is equal to n times n plus one over two.…
This course relies on an open-source SML (standard machine language) library to demo the concepts behind discrete math. Peggy Fisher shows you how to manipulate sets of data, write proofs and truth tables, analyze data sequences, and visualize data using graph theory. Challenges at the end of every chapter allow you to test your knowledge. By the end of the course, you should be able to make the leap from theory to using discrete math in practice: saving time and resulting in code that's cleaner and easier to maintain in the long run.
- Real-world discrete math
- Objects as sets
- Set notation and operations
- Standard machine language (SML) setup
- Working with data types, strings, and functions in SML
- Analyzing data sequences
- Writing truth tables
- Identifying and evaluating predicates
- Validating arguments
- Writing proofs: subset, conditional, and biconditional proofs
- Visualizing data with graphs
- Advanced discrete math techniques
Skill Level Intermediate
1. Discrete Math Uses
3. Setting Up SML
4. Analyzing Data Sequences
5. Effective Arguments and Defensible Decisions
6. Proofs Made Easy
7. Advanced Discrete Math Topics
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