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Discrete Mathematics

Discrete Mathematics. Section 5.02-Subsets ST 1. Objective: The students will…. Distinguish between a subset and a proper subset. Subsets. Large sets are often broken into smaller more manageable sets, called subsets.

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Discrete Mathematics

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  1. Discrete Mathematics Section 5.02-Subsets ST 1

  2. Objective: The students will… • Distinguish between a subset and a proper subset.

  3. Subsets Large sets are often broken into smaller more manageable sets, called subsets. • Subset: Set A is a subset of set B, symbolized by A B, if and only if all the elements of set A are also elements of set B.

  4. Subsets Example Set B is the people in your class. A subset of this could be any of the following: * females * males * students whose last name begins with M * students who are 17 year olds

  5. Example • Is N a subset of M? M = {5, 7, 9, 11, 13, 15}, N = {5, 7, 13} • Answer: Yes! Every element of set N is also found inside set M, so N M.

  6. You Try • Is set A a subset of set B? A = {2, 3, 4, 5}, B = {2, 3} • Answer: No! Set A has a 4 and a 5 that were not found inside set B, so A B. A is not a subset of B

  7. You Try • Write two sets, A and B, so that A B. • Possible Answer: A = {1, 2, 3} and B = {e, f, g} In this case: A B

  8. Proper Subset • Proper Subset: Set A is a proper subset of set B, symbolized by A B, if and only if all the elements of set A are elements of set B and Set A Set B (all subsets are proper subsets unless they are exactly the same)

  9. Proper Subset Example • Is set A a proper subset of set B if A={jazz, pop, hip hop}, B={classical, jazz, pop, rap, hip hop}? • Answer: Yes. Every element in Set A is found inside Set B but they are not identical, thus A B.

  10. Example • If M = {1, 2, 3, 4}, N = {2, 3, 4} and O = {1, 2, 3, 4}, are N and O proper subsets of M? • Answer: N is a proper subset of M, so N M • O is not a proper subset of M, so O M, but O M

  11. Important: Empty Set • The empty set is a proper subset of every set except itself. • Ex: If A = { } and B = { }, then A is NOT a proper subset of B (because they are identical) BUT: the empty set is a subset of every set including itself so, A B but A B

  12. Number of Distinct Subsets • The “number of distinct subsets” of a finite set A is , where n is the number of elements in set A.

  13. Distinct Subsets Example • Example: If A = {a, b, c} what is the number of distinct subsets? What are they? • Answer: so there are 8 subsets. The subsets are: {a, b, c}, {a, b}, {a, c}, {b, c}, {a}, {b}, {c}, { }

  14. You Try • At the local pizza shop, you can order pizza with pepperoni, sausage, Canadian bacon, olives, or anchovies. How many different types of pizza can be made? • Answer: types of pizza

  15. Lesson Reminders: • When dealing with identical sets, remember… • Every set is a subset of itself: A A • No set is a proper subset of itself: A A

  16. Assignment • Worksheet 5.02

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