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Set theory-complete-1211828121770367-8

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Set theory-complete-1211828121770367-8

1. 1. Set Theory
2. 2. Section 1Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of objects can be collected into a set, set theory is applied most often to objects that are relevant to mathematics.
3. 3. Elements/MembersIndividual objects contained in the collectionEx: {x,a,p, or d}
4. 4. Set-Builder NotationWe represent sets by listing elements or by using set-builder notationExample:C= {x : x is a carnivorous animal}
5. 5. Well DefinedA set is well defined if we are able to tell whether any particular object is an element of that set.A= {x : x is a winner of an Academy Award}T= {x : x is tall}
6. 6. Empty or Null SetThe set that contains no elements is called the empty set or null set. This is labeled by a symbol that has a 0 with a / going through it.
7. 7. Universal SetThe universal set is the set of all elements under consideration in a given discussion. It is often described by using the capital letter U.
8. 8. Cardinal NumbersThe actual number of elements in a Set is its cardinal number. It is described by using n(A).
9. 9. Finite and Infinite NumbersSet can either be finite or infinite depending on the whole number. If a sets cardinal number is a whole number then it is finite. If it is not, then it is infinite.
10. 10. 1.3 The Language of Sets Problems Use Set notation to list all the elements of each sets. M= The months of the year M= {January, February, March, April, May, June…} P=Pizza Toppings P={pepperoni, cheese, mushrooms, anchovies,…} Anyone ordered pizza?
11. 11. 1.3 The Language of Sets Problems Determine whether each set is Well Defined: {x:x lives in Michigan} Well Defined {y:y has an interesting job} Not Well Defined State Whether each set is finite or infinite. P={x:x is a planet in our solar system} Finite N={1,2,3,…} Infinite
12. 12. Equal SetsTwo sets can be considered equal if they have the exact same members in them. It would be written as A=B.If A and B were not equal then it would be A = B.
13. 13. SubsetA subset would occur if every element of one set is also an element in another set. Using A and B, we could say that all the elements of A were also in B too, and it would be wrote as A then a sideways U underlined with B after.
14. 14. Proper SubsetsUsing A and B, Set A would be a proper subset of B if A ¢ B but A = B.
15. 15. 1.3 The Language of Sets Problems 1) A={x : x lives in Raleigh} B={x : x lives in North Carolina} Is A a subset of B? Answer: Yes, A is a subset of B because, Raleigh lies within North Carolina 2) A={1,2,3} B={1,2,3,5,6,7) Is A a subset of B? Answer: Yes, A is a subset of B because the numbers in A are in B 3) A={1,2,3,4} B={1,2,3,5,6,7,8} Is A a subset of B? Answer: No, because 4 is not incuded in set B.
16. 16. UnionThe union of two sets would be wrote as A U B, which is the set of elements that are members of A or B, or both too.Using set-builder notation,A U B = {x : x is a member of A or X is a member of B}
17. 17. IntersectionIntersection are written as A ∩ B, is the set of elements that are in A and B.Using set-builder notation, it would look like:A ∩ B = {x : x is a member of A and x is a member of B}
18. 18. ComplementsWith A being a subset of the universal (U), the complement of A (A’) is the set of elements of U that are not elements of A.
19. 19. Other DefinitionsVenn diagram – a method of visualizing sets using various shapesDisjoint – If A ∩ B = 0, then A and B are disjoint.Difference: B – A; all the elements in B but not in AEquivalent sets – two sets are equivalent if n(A) = n(B).