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# Discreet_Set Theory

## by guest68d714 on May 26, 2008

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Burtion_Discreet_Set Theory

Burtion_Discreet_Set Theory

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## Discreet_Set TheoryPresentation Transcript

• Set Theory Jemel, Jenny, Ramon, Don, Irma
• 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.
• Elements/Members
• Individual objects contained in the collection
• Ex: { x,a,p, or d}
• Set-Builder Notation
• We represent sets by listing elements or by using set-builder notation
• Example:
• C= { x : x is a carnivorous animal}
• 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}
• 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.
• 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.
• Cardinal Numbers
• The actual number of elements in a Set is its cardinal number. It is described by using n(A).
• 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 .
• 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?
• 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
• 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.
• 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.
• Proper Subsets
• Using A and B, Set A would be a proper subset of B if A ¢ B but A = B.
• 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.
• 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}
• 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}
• 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.
• 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).