Join Peggy Fisher for an in-depth discussion in this video Write a general outline for a proof, part of Programming Foundations: Discrete Mathematics.
- A theorem is a statement that can be proven true or false.…A proof consists of a series of steps,…each of which follows logically from assumptions…or from previously proven statements…whose final step should result in a statement…of the theorem being proven.…Axioms are statements assumed to be true,…or previously proven theorems.…An example is the definition of even and odd integers.…An even integer is any integer…that can be represented by two times x,…where x is also an integer.…
10 is two times five,…200 is two times 100, etcetera.…An odd number is represented by any number…such that it can be written as two times x plus one.…So 11 is two times five, plus one.…Proofs fall into two categories, direct and indirect proofs.…With an indirect proof, instead of proving…that the conclusion is true, you start by assuming…that the opposite is true.…
Then, using deductive reasoning…to lead to a contradiction, you can prove…the original hypotheses is true.…Proof by contradiction is another type of indirect proof,…and requires a counter-example.…
Author
Peggy Fisher
Released
3/9/2016This 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
Duration
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Introduction
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Welcome1m 11s
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1. Discrete Math Uses
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Real-world discrete math3m 8s
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Abstract discrete math1m 57s
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2. Sets
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Objects as sets2m 56s
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Set notation3m 56s
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Set operations5m 1s
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Power sets4m 29s
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Sequences and sums7m 22s
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Recursion3m 5s
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Solution: Practice with sets6m 53s
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3. Setting Up SML
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Functional programming2m 31s
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Datatypes4m 45s
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Characters and strings5m 19s
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Recursive functions4m 45s
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Challenge: Learn SML1m 40s
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4. Analyzing Data Sequences
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Use SML to create lists4m 39s
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Perform functions on lists4m 31s
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5. Effective Arguments and Defensible Decisions
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Truth tables4m 58s
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Conditional propositions5m 48s
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Valid arguments4m 40s
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Rules of inference4m 45s
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Prove logical equivalence6m 11s
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Solution: Write truth tables4m 55s
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6. Proofs Made Easy
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Write subset proofs3m 12s
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Evaluate conditional proofs8m 54s
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Solution: Write a proof4m 23s
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7. Advanced Discrete Math Topics
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Event probability3m 31s
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Cryptography2m 22s
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Conclusion
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Next steps35s
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Video: Write a general outline for a proof