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Introduction to Sets (Optional)
Frank Ma © 2011
A set is a backpack which may or may not contains any items. Introduction to Sets
A set is a backpack which may or may not contains any items. We use capital letters as names of sets.
Introduction to Sets
A set is 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 setB = {wallet, my car–key, math–book} C = {my car–key}D = {1, 2, 3, ….} = {all positive integer}
Introduction to Sets
A set is 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 setB = {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 set is 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 setB = {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 set is 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 setB = {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 set is 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 setB = {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
Introduction to Sets
my car–keyϵ B
A set is 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 setB = {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
Introduction to Sets
my car–keyϵ B my car–keyϵ C
A set is 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 setB = {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
Introduction to Sets
my car–keyϵ B my car–keyϵ C my car–keyϵ A
A set is 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 setB = {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
Introduction to Sets
my car–keyϵ B my car–keyϵ C my car–keyϵ A
Note that C is not an element of B because C is just anotherbackpack whose content part (or all) of B’s content.
A set is 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 setB = {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
Introduction to Sets
my car–keyϵ B my car–keyϵ C my car–keyϵ A
Note that C is not an element of B because C is just anotherbackpack whose content part (or all) of B’s content. In generalT is a subset of S if every element of T is also an element of S and we write this as T S.
A set is 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 setB = {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
Introduction to Sets
my car–keyϵ B my car–keyϵ C my car–keyϵ A
Note that C is not an element of B because C is just anotherbackpack whose content part (or all) of B’s content. In generalT is a subset of S if every element of T is also an element of S and we write this as T S. So we’ve C but C B.ϵ B
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.
Introduction to Sets
Some sets of numbers may be represented graphically.
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
Some sets of numbers may be represented graphically.
Example A. Graph the following sets.a. A = {all the x where –2 < x ≤ 6}
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
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
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
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}
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
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}
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
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}
x–4 4
This is the same as { x where –4 ≤ x ≤ 4}
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
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}
x–4 4
This is the same as { x where –4 ≤ x ≤ 4}
Intersection and Union
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
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}
x–4 4
This is the same as { x where –4 ≤ x ≤ 4}
Intersection and UnionLet S and T be two sets, S T, read as S intersects T, is the set of common elements of S and T.
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
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}
x–4 4
This is the same as { x where –4 ≤ x ≤ 4}
Intersection and UnionLet 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}.
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
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}
x–4 4
This is the same as { x where –4 ≤ x ≤ 4}
Intersection and UnionLet 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 SetsLet S and T be two sets, S U T , read as S unions T, is the merge–set of the two sets.
Introduction to SetsLet 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 SetsLet 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
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}
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
Example B. Find and draw A B and A U B given that
A = {all the x where –2 < x ≤ 6}
x–2 6
B = {all the x where | x | ≤ 4}
x–4 4
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}.
A
B
We have graphically
Introduction to Sets
Example B. Find and draw A B and A U B given that
A = {all the x where –2 < x ≤ 6}
x–2 6
B = {all the x where | x | ≤ 4}
x–4 4
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}.
A
B
We have graphically
A B is the common or overlapped segment of A with B.–2 4
A B
Introduction to Sets
Example B. Find and draw A B and A U B given that
A = {all the x where –2 < x ≤ 6}
x–2 6
B = {all the x where | x | ≤ 4}
x–4 4
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}.
A
B
We have graphically
A B is the common or overlapped segment of A with B. A = { B = {–2 < x ≤ 4}
–2 4
A B
Introduction to Sets
x–2 6
x–4 4
A
B
Given that
Introduction to Sets
x–2 6
x–4 4
A
B
A U B consists both portions of either color
Given that
Introduction to Sets
x–2 6
x–4 4
A
B
A U B consists both portions of either color = {–4 < x ≤ 6}.6–4
A U B
Given that
Introduction to Sets
x–2 6
x–4 4
A
B
A U B consists both portions of either color = {–4 < x ≤ 6}.6–4
A U B
Given that
Let’s extend this to the x&y coordinate system.
Introduction to Sets
x–2 6
x–4 4
A
B
A U B consists both portions of either color = {–4 < x ≤ 6}.6–4
A U B
Given that
Let’s extend this to the x&y coordinate system. The basic 2D regions are half–planes and strips.
Introduction to Sets
x–2 6
x–4 4
A
B
A U B consists both portions of either color = {–4 < x ≤ 6}.6–4
A U B
Given that
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
x–2 6
x–4 4
A
B
A U B consists both portions of either color = {–4 < x ≤ 6}.6–4
A U B
Given that
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
x–2 6
x–4 4
A
B
A U B consists both portions of either color = {–4 < x ≤ 6}.6–4
A U B
Given that
Let’s extend this to the x&y coordinate system. The basic 2D regions are half–planes and strips.
{(x, y) where x > 0}
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
x–2 6
x–4 4
A
B
A U B consists both portions of either color = {–4 < x ≤ 6}.6–4
A U B
Given that
Let’s extend this to the x&y coordinate system. The basic 2D regions are half–planes and strips.
{(x, y) where x > 0}
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. The open region is called a half–plane and the dash–line means exclusion.
Introduction to Setsb. B = {(x, y) where 0 < y ≤ 4}
Introduction to Setsb. B = {(x, y) where 0 < y ≤ 4}
There is no restriction on x so x may take on any value.
Introduction to Setsb. 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 Setsb. 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}
The open region we obtained is called a strip.
Introduction to Setsb. 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}
The open region we obtained is called a strip. Note that the solid represent inclusion.
Introduction to Setsb. 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}
The open region we obtained is called a strip. Note that the solid represent inclusion.
c. Let C = {(x, y) where –3 < x < 3 and 0 < y ≤ 4} draw B C.
Introduction to Setsb. 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}
The open region we obtained is called a strip. Note that the solid represent inclusion.
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 Setsb. 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}
The open region we obtained is called a strip. Note that the solid represent inclusion.
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 Setsb. 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}
The open region we obtained is called a strip. Note that the solid represent inclusion.
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.
BC is the vertical strip where–3 < x < 3.
C
Introduction to Setsb. 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}
The open region we obtained is called a strip. Note that the solid represent inclusion.
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.
BC is the vertical strip where–3 < x < 3. The overlap is the rectangular region. B C
C
Introduction to Setsb. 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}
The open region we obtained is called a strip. Note that the solid represent inclusion.
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.C is the vertical strip where–3 < x < 3. The overlap is the rectangular region. Note that all the cornersare excluded.
B
B C
C
Introduction to Setsb. 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}
The open region we obtained is called a strip. Note that the solid represent inclusion.
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.C is the vertical strip where–3 < x < 3. The overlap is the rectangular region. Note that all the cornersare excluded. You Do: Label the corners.
B
C
B C
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