3 6 introduction to sets-optional

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

3 6 introduction to sets-optional

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