Join Peggy Fisher for an in-depth discussion in this video Sets from Cartesian products, part of Programming Foundations: Discrete Mathematics.
- [Voiceover] Creating sets from Cartesian products.…Cartesian products are a result…of taking two sets, such as sets A and B,…and creating a list of ordered pairs.…A more formal definition looks like this.…A cross B is equal to the ordered pairs of…a comma b, such that…little A is an element of set A,…and little B is an element of set B.…Using this formula, if we have sets A…a set of all students at a college,…and B, a set of all courses at the college,…A times B would be the set of…all student and course combinations.…
To predetermine the number of ordered pairs…we can take the cardinality of set A…times the cardinality of set B.…That gives us the number of ordered pairs.…In this example, we will keep it simple…and say that we have two students…and three possible courses.…The cardinality of A is two,…the cardinality of B is three,…so we will end up with six ordered pairs.…To find the Cartesian product of these two sets…we start by taking a value from the first set, such as s1,…and we combine it with every element in the second set.…
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: Sets from Cartesian products