Discreet_Set Theory


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Discreet_Set Theory

  1. 1. Set Theory Jemel, Jenny, Ramon, Don, Irma
  2. 2. Section 1 <ul><li>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. </li></ul>
  3. 3. Elements/Members <ul><li>Individual objects contained in the collection </li></ul><ul><li>Ex: { x,a,p, or d} </li></ul>
  4. 4. Set-Builder Notation <ul><li>We represent sets by listing elements or by using set-builder notation </li></ul><ul><li>Example: </li></ul><ul><li>C= { x : x is a carnivorous animal} </li></ul>
  5. 5. Well Defined <ul><li>A set is well defined if we are able to tell whether any particular object is an element of that set. </li></ul><ul><li>A= { x : x is a winner of an Academy Award} </li></ul><ul><li>T= { x : x is tall} </li></ul>
  6. 6. Empty or Null Set <ul><li>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. </li></ul>
  7. 7. Universal Set <ul><li>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. </li></ul>
  8. 8. Cardinal Numbers <ul><li>The actual number of elements in a Set is its cardinal number. It is described by using n(A). </li></ul>
  9. 9. Finite and Infinite Numbers <ul><li>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 . </li></ul>
  10. 10. 1.3 The Language of Sets Problems <ul><li>Use Set notation to list all the elements of each sets. </li></ul><ul><li>M= The months of the year </li></ul><ul><ul><li>M= {January, February, March, April, May, June…} </li></ul></ul><ul><li>P=Pizza Toppings </li></ul><ul><ul><li>P={pepperoni, cheese, mushrooms, anchovies,…} </li></ul></ul>Anyone ordered pizza?
  11. 11. 1.3 The Language of Sets Problems <ul><li>Determine whether each set is Well Defined: </li></ul><ul><li>{x:x lives in Michigan} </li></ul><ul><ul><li>Well Defined </li></ul></ul><ul><li>{y:y has an interesting job} </li></ul><ul><ul><li>Not Well Defined </li></ul></ul><ul><li>State Whether each set is finite or infinite. </li></ul><ul><li>P={x:x is a planet in our solar system} </li></ul><ul><ul><li>Finite </li></ul></ul><ul><li>N={1,2,3,…} </li></ul><ul><ul><li>Infinite </li></ul></ul>
  12. 12. Equal Sets <ul><li>Two sets can be considered equal if they have the exact same members in them. It would be written as A=B. </li></ul><ul><li>If A and B were not equal then it would be A = B. </li></ul>
  13. 13. Subset <ul><li>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. </li></ul>
  14. 14. Proper Subsets <ul><li>Using A and B, Set A would be a proper subset of B if A ¢ B but A = B. </li></ul>
  15. 15. 1.3 The Language of Sets Problems <ul><li>1) A={x : x lives in Raleigh} B={x : x lives in North Carolina} </li></ul><ul><li>Is A a subset of B? </li></ul><ul><ul><li>Answer: Yes, A is a subset of B because, Raleigh lies within North Carolina </li></ul></ul><ul><li>2) A={1,2,3} B={1,2,3,5,6,7) </li></ul><ul><li>Is A a subset of B? </li></ul><ul><ul><li>Answer: Yes, A is a subset of B because the numbers in A are in B </li></ul></ul><ul><li>3) A={1,2,3,4} B={1,2,3,5,6,7,8} </li></ul><ul><li>Is A a subset of B? </li></ul><ul><ul><li>Answer: No, because 4 is not incuded in set B. </li></ul></ul>
  16. 16. Union <ul><li>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. </li></ul><ul><li>Using set-builder notation, </li></ul><ul><li>A U B = {x : x is a member of A or X is a member of B} </li></ul>
  17. 17. Intersection <ul><li>Intersection are written as A ∩ B, is the set of elements that are in A and B. </li></ul><ul><li>Using set-builder notation, it would look like: </li></ul><ul><li>A ∩ B = {x : x is a member of A and x is a member of B} </li></ul>
  18. 18. Complements <ul><li>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. </li></ul>
  19. 19. Other Definitions <ul><li>Venn diagram – a method of visualizing sets using various shapes </li></ul><ul><li>Disjoint – If A ∩ B = 0, then A and B are disjoint. </li></ul><ul><li>Difference: B – A; all the elements in B but not in A </li></ul><ul><li>Equivalent sets – two sets are equivalent if n(A) = n(B). </li></ul>