Peer Instruction in Discrete Mathematics
Explore topics in discrete mathematics such as set sizes, set builder notation, power sets, Cartesian products, unions, intersections, and different ways of defining sets. Learn through engaging visuals and examples presented under a Creative Commons License by Dr. Cynthia Bailey Lee and Dr. Shachar Lovett.
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Creative Commons License CSE 20 Discrete Mathematics Dr. Cynthia Bailey Lee Dr. Shachar Lovett Peer Instruction in Discrete Mathematics by Cynthia Leeis licensed under a Creative Commons Attribution- NonCommercial-ShareAlike 4.0 International License. Based on a work at http://peerinstruction4cs.org. Permissions beyond the scope of this license may be available at http://peerinstruction4cs.org.
2 Today s Topics: 1. Set sizes 2. Set builder notation 3. Rapid-fire set-theory practice
3 1. Set sizes
4 Power set Let A be a set of n elements (|A|=n) How large is P(A) (the power-set of A)? A. n B. 2n C. n2 D. 2n E. None/other/more than one
5 Cartesian product |A|=n, |B|=m How large is A x B ? A. n+m B. nm C. n2 D. m2 E. None/other/more than one
6 Union |A|=n, |B|=m How large is A B ? A. n+m B. nm C. n2 D. m2 E. None/other/more than one
7 Intersection |A|=n, |B|=m How large is A B ? A. n+m B. nm C. At most n D. At most m E. None/other/more than one
8 2. Set builder notation
9 Set builder notation Example: Even Our definition of Even is: ? ?,? = 2?. How can we write this as a set, rather than a definition applying to an individual n? n ? ? ?,? = 2?}, or just n ? n ? ?,? = 2?} | is pronounced such that
10 Set builder notation How could we write the set of integers that are multiples of 12 ? A. { ? ?,? ?,? = 3? ??? n = 4j} ? ? ?,? = 3? ??? ? ?,n = 4j} { 36, 24, 12,0,12,24,36 } ? ? ? ?,? = 12?} E. Other/none/more than one B. C. D.
11 Ways of defining a set Enumeration: {1,2,3,4,5,6,7,8,9} + very clear - impractical for large sets Incomplete enumeration (ellipses): {1,2,3, ,98,99,100} + takes up less space, can work for large or infinite sets - not always clear {2 3 5 7 11 13 } What does this mean? What is the next element? Set builder: { n | <some criteria>} + can be used for large or infinite sets, clearly sets forth rules for membership
12 Primes Enumeration may not be clear: {2 3 5 7 11 13 } How can we write the set Primes using set builder notation?
13 3. Rapid-fire set-theory practice Clickers ready!
14 Set Theory rapid-fire practice (A and B are sets) ? ?,B ?. A. TRUE B. FALSE In your discussion: If true, prove it! (quickly sketch out what the argument would be) If false, what are the counterexample A and B?
15 Set Theory rapid-fire practice (A and B are sets) ? , ?,B ?. A. TRUE B. FALSE In your discussion: If true, prove it! (quickly sketch out what the argument would be) If false, what are the counterexample A and B?
16 Set Theory rapid-fire practice (A is a set) ?, ? A. TRUE B. FALSE In your discussion: If true, prove it! (quickly sketch out what the argument would be) If false, what are the counterexample A and B?
