Join Peggy Fisher for an in-depth discussion in this video Solution: Write a proof, part of Programming Foundations: Discrete Mathematics.
- [Voiceover] How did you make out…with the proof challenges?…I'd like to go over my solution.…The first proof said if three x plus five…is an odd number, then x has to be an even integer.…So for this example, we let x equal two k…which our definition of even numbers.…We substitute two k in for x…and we get three times two k plus five…which is six k plus five.…Now here's the tricky part.…Remember five is the same thing as four plus one.…
So I'm gonna replace five…and I get six k plus four plus one.…Now I can factor it two out of the six k plus four…and now I get two times a quantity…three k plus two plus one.…And remember, integers are close under addition.…So that means three k plus two is an integer…and two times an integer plus one is an odd number.…So the original proof holds.…The next proof says, "If the square of an integer x is even,…"then x is even."…This one might be easier to approach…using the contrapositive.…
Remember the contrapositive is…logically equivalent to the original proof.…So we start by saying assume x is odd.…
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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