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3 6 introduction to sets-optional

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    3 6 introduction to sets-optional 3 6 introduction to sets-optional Presentation Transcript

    • Introduction to Sets
      (Optional)
      Frank Ma © 2011
    • Introduction to Sets
      A setis a backpack which may or may not contains any items.
    • Introduction to Sets
      A setis a backpack which may or may not contains any items.
      We use capital letters as names of sets.
    • Introduction to Sets
      A setis a backpack which may or may not contains any items.
      We use capital letters as names of sets. For example,
      A = { } ≡ Ф (phi) – which is the empty set
      B = {wallet, my car–key, math–book}
      C = {my car–key}
      D = {1, 2, 3, ….} = {all positive integer}
    • Introduction to Sets
      A setis a backpack which may or may not contains any items.
      We use capital letters as names of sets. For example,
      A = { } ≡ Ф (phi) – which is the empty set
      B = {wallet, my car–key, math–book}
      C = {my car–key}
      D = {1, 2, 3, ….} = {all positive integer}
      Note that we use “{ }” to symbolize the backpack and also that a set may contain infinitely many items.
    • Introduction to Sets
      A setis a backpack which may or may not contains any items.
      We use capital letters as names of sets. For example,
      A = { } ≡ Ф (phi) – which is the empty set
      B = {wallet, my car–key, math–book}
      C = {my car–key}
      D = {1, 2, 3, ….} = {all positive integer}
      Note that we use “{ }” to symbolize the backpack and also that a set may contain infinitely many items.
      Each item of a set is called an element of that set.
    • Introduction to Sets
      A setis a backpack which may or may not contains any items.
      We use capital letters as names of sets. For example,
      A = { } ≡ Ф (phi) – which is the empty set
      B = {wallet, my car–key, math–book}
      C = {my car–key}
      D = {1, 2, 3, ….} = {all positive integer}
      Note that we use “{ }” to symbolize the backpack and also that a set may contain infinitely many items.
      Each item of a set is called an element of that set.
      We write x ϵ S if x is an element the set S.
    • Introduction to Sets
      A setis a backpack which may or may not contains any items.
      We use capital letters as names of sets. For example,
      A = { } ≡ Ф (phi) – which is the empty set
      B = {wallet, my car–key, math–book}
      C = {my car–key}
      D = {1, 2, 3, ….} = {all positive integer}
      Note that we use “{ }” to symbolize the backpack and also that a set may contain infinitely many items.
      Each item of a set is called an element of that set.
      We write x ϵ S if x is an element the set S.
      Hence from the above sets we’ve that
      my car–key
      ϵ B
    • Introduction to Sets
      A setis a backpack which may or may not contains any items.
      We use capital letters as names of sets. For example,
      A = { } ≡ Ф (phi) – which is the empty set
      B = {wallet, my car–key, math–book}
      C = {my car–key}
      D = {1, 2, 3, ….} = {all positive integer}
      Note that we use “{ }” to symbolize the backpack and also that a set may contain infinitely many items.
      Each item of a set is called an element of that set.
      We write x ϵ S if x is an element the set S.
      Hence from the above sets we’ve that
      my car–key
      ϵ B
      my car–key
      ϵ C
    • Introduction to Sets
      A setis a backpack which may or may not contains any items.
      We use capital letters as names of sets. For example,
      A = { } ≡ Ф (phi) – which is the empty set
      B = {wallet, my car–key, math–book}
      C = {my car–key}
      D = {1, 2, 3, ….} = {all positive integer}
      Note that we use “{ }” to symbolize the backpack and also that a set may contain infinitely many items.
      Each item of a set is called an element of that set.
      We write x ϵ S if x is an element the set S.
      Hence from the above sets we’ve that
      my car–key
      ϵ A
      my car–key
      ϵ B
      my car–key
      ϵ C
    • Introduction to Sets
      A setis a backpack which may or may not contains any items.
      We use capital letters as names of sets. For example,
      A = { } ≡ Ф (phi) – which is the empty set
      B = {wallet, my car–key, math–book}
      C = {my car–key}
      D = {1, 2, 3, ….} = {all positive integer}
      Note that we use “{ }” to symbolize the backpack and also that a set may contain infinitely many items.
      Each item of a set is called an element of that set.
      We write x ϵ S if x is an element the set S.
      Hence from the above sets we’ve that
      my car–key
      ϵ A
      my car–key
      ϵ B
      my car–key
      ϵ C
      Note that C is not an element of B because C is just another
      backpack whose content part (or all) of B’s content.
    • Introduction to Sets
      A setis a backpack which may or may not contains any items.
      We use capital letters as names of sets. For example,
      A = { } ≡ Ф (phi) – which is the empty set
      B = {wallet, my car–key, math–book}
      C = {my car–key}
      D = {1, 2, 3, ….} = {all positive integer}
      Note that we use “{ }” to symbolize the backpack and also that a set may contain infinitely many items.
      Each item of a set is called an element of that set.
      We write x ϵ S if x is an element the set S.
      Hence from the above sets we’ve that
      my car–key
      ϵ A
      my car–key
      ϵ B
      my car–key
      ϵ C
      Note that C is not an element of B because C is just another
      backpack whose content part (or all) of B’s content. In general
      T is a subset of S if every element of T is also an element of S and we write this asT S.
    • Introduction to Sets
      A setis a backpack which may or may not contains any items.
      We use capital letters as names of sets. For example,
      A = { } ≡ Ф (phi) – which is the empty set
      B = {wallet, my car–key, math–book}
      C = {my car–key}
      D = {1, 2, 3, ….} = {all positive integer}
      Note that we use “{ }” to symbolize the backpack and also that a set may contain infinitely many items.
      Each item of a set is called an element of that set.
      We write x ϵ S if x is an element the set S.
      Hence from the above sets we’ve that
      my car–key
      ϵ A
      my car–key
      ϵ B
      my car–key
      ϵ C
      Note that C is not an element of B because C is just another
      backpack whose content part (or all) of B’s content. In general
      T is a subset of S if every element of T is also an element of S and we write this asT S. So we’ve C but C B.
      ϵ B
    • Introduction to Sets
      Two sets S and T are the same if they have exactly the same content, that is, every element in S is also an element of T and vice versa.
    • Introduction to Sets
      Two sets S and T are the same if they have exactly the same content, that is, every element in S is also an element of T and vice versa. Hence if S = {a, b} and that T = {b ,a} then S = T.
    • Introduction to Sets
      Two sets S and T are the same if they have exactly the same content, that is, every element in S is also an element of T and vice versa. Hence if S = {a, b} and that T = {b ,a} then S = T.
      Note that the order of the elements is not important.
    • Introduction to Sets
      Two sets S and T are the same if they have exactly the same content, that is, every element in S is also an element of T and vice versa. Hence if S = {a, b} and that T = {b ,a} then S = T.
      Note that the order of the elements is not important.
      Some sets of numbers may be represented graphically.
    • Introduction to Sets
      Two sets S and T are the same if they have exactly the same content, that is, every element in S is also an element of T and vice versa. Hence if S = {a, b} and that T = {b ,a} then S = T.
      Note that the order of the elements is not important.
      Some sets of numbers may be represented graphically.
      Example A. Graph the following sets.
      a. A = {all the x where –2 < x ≤ 6}
    • Introduction to Sets
      Two sets S and T are the same if they have exactly the same content, that is, every element in S is also an element of T and vice versa. Hence if S = {a, b} and that T = {b ,a} then S = T.
      Note that the order of the elements is not important.
      Some sets of numbers may be represented graphically.
      Example A. Graph the following sets.
      a. A = {all the x where –2 < x ≤ 6}
      x
      –2
      6
    • Introduction to Sets
      Two sets S and T are the same if they have exactly the same content, that is, every element in S is also an element of T and vice versa. Hence if S = {a, b} and that T = {b ,a} then S = T.
      Note that the order of the elements is not important.
      Some sets of numbers may be represented graphically.
      Example A. Graph the following sets.
      a. A = {all the x where –2 < x ≤ 6}
      x
      –2
      6
      b. B = {all the x where | x | ≤ 4}
    • Introduction to Sets
      Two sets S and T are the same if they have exactly the same content, that is, every element in S is also an element of T and vice versa. Hence if S = {a, b} and that T = {b ,a} then S = T.
      Note that the order of the elements is not important.
      Some sets of numbers may be represented graphically.
      Example A. Graph the following sets.
      a. A = {all the x where –2 < x ≤ 6}
      x
      –2
      6
      b. B = {all the x where | x | ≤ 4}
      This is the same as { x where –4 ≤ x ≤ 4}
    • Introduction to Sets
      Two sets S and T are the same if they have exactly the same content, that is, every element in S is also an element of T and vice versa. Hence if S = {a, b} and that T = {b ,a} then S = T.
      Note that the order of the elements is not important.
      Some sets of numbers may be represented graphically.
      Example A. Graph the following sets.
      a. A = {all the x where –2 < x ≤ 6}
      x
      –2
      6
      b. B = {all the x where | x | ≤ 4}
      This is the same as { x where –4 ≤ x ≤ 4}
      x
      –4
      4
    • Introduction to Sets
      Two sets S and T are the same if they have exactly the same content, that is, every element in S is also an element of T and vice versa. Hence if S = {a, b} and that T = {b ,a} then S = T.
      Note that the order of the elements is not important.
      Some sets of numbers may be represented graphically.
      Example A. Graph the following sets.
      a. A = {all the x where –2 < x ≤ 6}
      x
      –2
      6
      b. B = {all the x where | x | ≤ 4}
      This is the same as { x where –4 ≤ x ≤ 4}
      x
      –4
      4
      Intersection and Union
    • Introduction to Sets
      Two sets S and T are the same if they have exactly the same content, that is, every element in S is also an element of T and vice versa. Hence if S = {a, b} and that T = {b ,a} then S = T.
      Note that the order of the elements is not important.
      Some sets of numbers may be represented graphically.
      Example A. Graph the following sets.
      a. A = {all the x where –2 < x ≤ 6}
      x
      –2
      6
      b. B = {all the x where | x | ≤ 4}
      This is the same as { x where –4 ≤ x ≤ 4}
      x
      –4
      4
      Intersection and Union
      Let S and T be two sets, S T, read as S intersects T, is the set of common elements of S and T.
    • Introduction to Sets
      Two sets S and T are the same if they have exactly the same content, that is, every element in S is also an element of T and vice versa. Hence if S = {a, b} and that T = {b ,a} then S = T.
      Note that the order of the elements is not important.
      Some sets of numbers may be represented graphically.
      Example A. Graph the following sets.
      a. A = {all the x where –2 < x ≤ 6}
      x
      –2
      6
      b. B = {all the x where | x | ≤ 4}
      This is the same as { x where –4 ≤ x ≤ 4}
      x
      –4
      4
      Intersection and Union
      Let S and T be two sets, S T, read as S intersects T, is the set of common elements of S and T. Hence {a, b} {b, c} = {b}.
    • Introduction to Sets
      Two sets S and T are the same if they have exactly the same content, that is, every element in S is also an element of T and vice versa. Hence if S = {a, b} and that T = {b ,a} then S = T.
      Note that the order of the elements is not important.
      Some sets of numbers may be represented graphically.
      Example A. Graph the following sets.
      a. A = {all the x where –2 < x ≤ 6}
      x
      –2
      6
      b. B = {all the x where | x | ≤ 4}
      This is the same as { x where –4 ≤ x ≤ 4}
      x
      –4
      4
      Intersection and Union
      Let S and T be two sets, S T, read as S intersects T, is the set of common elements of S and T. Hence {a, b} {b, c} = {b}.
      Note that the intersection is a set, not just the element “b”.
    • Introduction to Sets
      Let S and T be two sets, S U T , read as S unions T, is the merge–set of the two sets.
    • Introduction to Sets
      Let S and T be two sets, S U T , read as S unions T, is the merge–set of the two sets. For the union operation, duplicated items is only counted once.
    • Introduction to Sets
      Let S and T be two sets, S U T , read as S unions T, is the merge–set of the two sets. For the union operation, duplicated items is only counted once. Hence {a, b} U {b, c} = {a, b, c}.
    • Introduction to Sets
      Let S and T be two sets, S U T , read as S unions T, is the merge–set of the two sets. For the union operation, duplicated items is only counted once. Hence {a, b} U {b, c} = {a, b, c}.
      Example B. Find and draw A B and A U B given that
      A = {all the x where –2 < x ≤ 6}
      B = {all the x where | x | ≤ 4}
    • Introduction to Sets
      Let S and T be two sets, S U T , read as S unions T, is the merge–set of the two sets. For the union operation, duplicated items is only counted once. Hence {a, b} U {b, c} = {a, b, c}.
      Example B. Find and draw A B and A U B given that
      A = {all the x where –2 < x ≤ 6}
      B = {all the x where | x | ≤ 4}
      We have graphically
      A
      x
      –4
      4
      B
      x
      –2
      6
    • Introduction to Sets
      Let S and T be two sets, S U T , read as S unions T, is the merge–set of the two sets. For the union operation, duplicated items is only counted once. Hence {a, b} U {b, c} = {a, b, c}.
      Example B. Find and draw A B and A U B given that
      A = {all the x where –2 < x ≤ 6}
      B = {all the x where | x | ≤ 4}
      We have graphically
      A
      x
      –4
      4
      B
      x
      –2
      6
      A B
      4
      –2
      A B is the common or overlapped segment of A with B.
    • Introduction to Sets
      Let S and T be two sets, S U T , read as S unions T, is the merge–set of the two sets. For the union operation, duplicated items is only counted once. Hence {a, b} U {b, c} = {a, b, c}.
      Example B. Find and draw A B and A U B given that
      A = {all the x where –2 < x ≤ 6}
      B = {all the x where | x | ≤ 4}
      We have graphically
      A
      x
      –4
      4
      B
      x
      –2
      6
      A B
      4
      –2
      A B is the common or overlapped segment of A with B.
      A = { B = {–2 < x ≤ 4}
    • Introduction to Sets
      Given that
      A
      x
      –4
      4
      B
      x
      –2
      6
    • Introduction to Sets
      Given that
      A
      x
      –4
      4
      B
      x
      –2
      6
      A U B consists both portions of either color
    • Introduction to Sets
      Given that
      A
      x
      –4
      4
      B
      x
      –2
      6
      A U B
      –4
      6
      A U B consists both portions of either color = {–4 < x ≤ 6}.
    • Introduction to Sets
      Given that
      A
      x
      –4
      4
      B
      x
      –2
      6
      A U B
      –4
      6
      A U B consists both portions of either color = {–4 < x ≤ 6}.
      Let’s extend this to the x&y coordinate system.
    • Introduction to Sets
      Given that
      A
      x
      –4
      4
      B
      x
      –2
      6
      A U B
      –4
      6
      A U B consists both portions of either color = {–4 < x ≤ 6}.
      Let’s extend this to the x&y coordinate system. The basic 2D regions are half–planes and strips.
    • Introduction to Sets
      Given that
      A
      x
      –4
      4
      B
      x
      –2
      6
      A U B
      –4
      6
      A U B consists both portions of either color = {–4 < x ≤ 6}.
      Let’s extend this to the x&y coordinate system. The basic 2D regions are half–planes and strips.
      Example. C. Shade the following sets in the rectangular system.
      a. A = {(x, y) where x > 0}
    • Introduction to Sets
      Given that
      A
      x
      –4
      4
      B
      x
      –2
      6
      A U B
      –4
      6
      A U B consists both portions of either color = {–4 < x ≤ 6}.
      Let’s extend this to the x&y coordinate system. The basic 2D regions are half–planes and strips.
      Example. C. Shade the following sets in the rectangular system.
      a. A = {(x, y) where x > 0}
      Note that there is no mention of y means that y may take on any value.
    • Introduction to Sets
      Given that
      A
      x
      –4
      4
      B
      x
      –2
      6
      A U B
      –4
      6
      A U B consists both portions of either color = {–4 < x ≤ 6}.
      Let’s extend this to the x&y coordinate system. The basic 2D regions are half–planes and strips.
      Example. C. Shade the following sets in the rectangular system.
      a. A = {(x, y) where x > 0}
      {(x, y) where x > 0}
      Note that there is no mention of y means that y may take on any value.
    • Introduction to Sets
      Given that
      A
      x
      –4
      4
      B
      x
      –2
      6
      A U B
      –4
      6
      A U B consists both portions of either color = {–4 < x ≤ 6}.
      Let’s extend this to the x&y coordinate system. The basic 2D regions are half–planes and strips.
      Example. C. Shade the following sets in the rectangular system.
      a. A = {(x, y) where x > 0}
      {(x, y) where x > 0}
      Note that there is no mention of y means that y may take on any value. The open region is called ahalf–plane and the dash–line means exclusion.
    • Introduction to Sets
      b. B = {(x, y) where 0 < y ≤ 4}
    • Introduction to Sets
      b. B = {(x, y) where 0 < y ≤ 4}
      There is no restriction on x
      so x may take on any value.
    • Introduction to Sets
      b. B = {(x, y) where 0 < y ≤ 4}
      There is no restriction on x
      so x may take on any value.
      {(x, y) where 0 < y ≤ 4}
    • Introduction to Sets
      b. B = {(x, y) where 0 < y ≤ 4}
      There is no restriction on x
      so x may take on any value.
      The open region we obtained is called a strip.
      {(x, y) where 0 < y ≤ 4}
    • Introduction to Sets
      b. B = {(x, y) where 0 < y ≤ 4}
      There is no restriction on x
      so x may take on any value.
      The open region we obtained is called a strip. Note that the solid represent inclusion.
      {(x, y) where 0 < y ≤ 4}
    • Introduction to Sets
      b. B = {(x, y) where 0 < y ≤ 4}
      There is no restriction on x
      so x may take on any value.
      The open region we obtained is called a strip. Note that the solid represent inclusion.
      {(x, y) where 0 < y ≤ 4}
      c. Let C = {(x, y) where –3 < x < 3 and 0 < y ≤ 4}
      draw B C.
    • Introduction to Sets
      b. B = {(x, y) where 0 < y ≤ 4}
      There is no restriction on x
      so x may take on any value.
      The open region we obtained is called a strip. Note that the solid represent inclusion.
      {(x, y) where 0 < y ≤ 4}
      c. Let C = {(x, y) where –3 < x < 3 and 0 < y ≤ 4}
      draw B C.
      This is the overlap of two strips.
    • Introduction to Sets
      b. B = {(x, y) where 0 < y ≤ 4}
      There is no restriction on x
      so x may take on any value.
      The open region we obtained is called a strip. Note that the solid represent inclusion.
      {(x, y) where 0 < y ≤ 4}
      c. Let C = {(x, y) where –3 < x < 3 and 0 < y ≤ 4}
      draw B C.
      This is the overlap of two strips.
      B is the same as the above.
      B
    • Introduction to Sets
      b. B = {(x, y) where 0 < y ≤ 4}
      There is no restriction on x
      so x may take on any value.
      The open region we obtained is called a strip. Note that the solid represent inclusion.
      {(x, y) where 0 < y ≤ 4}
      c. Let C = {(x, y) where –3 < x < 3 and 0 < y ≤ 4}
      draw B C.
      C
      This is the overlap of two strips.
      B is the same as the above.
      C is the vertical strip where
      –3 < x < 3.
      B
    • Introduction to Sets
      b. B = {(x, y) where 0 < y ≤ 4}
      There is no restriction on x
      so x may take on any value.
      The open region we obtained is called a strip. Note that the solid represent inclusion.
      {(x, y) where 0 < y ≤ 4}
      c. Let C = {(x, y) where –3 < x < 3 and 0 < y ≤ 4}
      draw B C.
      C
      This is the overlap of two strips.
      B is the same as the above.
      C is the vertical strip where
      –3 < x < 3.
      The overlap is the rectangular region.
      B
      B C
    • Introduction to Sets
      b. B = {(x, y) where 0 < y ≤ 4}
      There is no restriction on x
      so x may take on any value.
      The open region we obtained is called a strip. Note that the solid represent inclusion.
      {(x, y) where 0 < y ≤ 4}
      c. Let C = {(x, y) where –3 < x < 3 and 0 < y ≤ 4}
      draw B C.
      C
      This is the overlap of two strips.
      B is the same as the above.
      C is the vertical strip where
      –3 < x < 3.
      The overlap is the rectangular region. Note that all the corners
      are excluded.
      B
      B C
    • Introduction to Sets
      b. B = {(x, y) where 0 < y ≤ 4}
      There is no restriction on x
      so x may take on any value.
      The open region we obtained is called a strip. Note that the solid represent inclusion.
      {(x, y) where 0 < y ≤ 4}
      c. Let C = {(x, y) where –3 < x < 3 and 0 < y ≤ 4}
      draw B C.
      C
      This is the overlap of two strips.
      B is the same as the above.
      C is the vertical strip where
      –3 < x < 3.
      The overlap is the rectangular region. Note that all the corners
      are excluded.
      You Do: Label the corners.
      B
      B C