i want to know about the chapter sets
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set is not soo imp. but the last topic 1.6 exercise is a little bit imp. otherwise it is not soo imp. chaper
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hey mate here is ur answer
In mathematics, a set is a collection of distinct objects, considered as an object in its own right. For example, the numbers 2, 4, and 6 are distinct objects when considered separately, but when they are considered collectively they form a single set of size three, written {2, 4, 6}. The concept of a set is one of the most fundamental in mathematics. Developed at the end of the 19th century, set theory is now a ubiquitous part of mathematics, and can be used as a foundation from which nearly all of mathematics can be derived. In mathematics education, elementary topics from set theory such as Venn diagrams are taught at a young age, while more advanced concepts are taught as part of a university degree.
There are several fundamental operations for constructing new sets from given sets.
UnionsEdit
The union of A and B, denoted A ∪ B
Main article: Union (set theory)
Two sets can be "added" together. The unionof A and B, denoted by A ∪ B, is the set of all things that are members of either A or B.
Examples:
{1, 2} ∪ {1, 2} = {1, 2}.{1, 2} ∪ {2, 3} = {1, 2, 3}.{1, 2, 3} ∪ {3, 4, 5} = {1, 2, 3, 4, 5}
Some basic properties of unions:
A ∪ B = B ∪ A.A ∪ (B ∪ C) = (A ∪ B) ∪ C.A ⊆ (A ∪ B).A ∪ A = A.A ∪ U = U.A ∪ ∅ = A.A ⊆ B if and only if A ∪ B = B.IntersectionsEdit
Main article: Intersection (set theory)
A new set can also be constructed by determining which members two sets have "in common". The intersection of A and B, denoted by A ∩ B, is the set of all things that are members of both A and B. If A ∩ B = ∅,then A and B are said to be disjoint.
The intersection of A and B, denoted A ∩ B.
Examples:
{1, 2} ∩ {1, 2} = {1, 2}.{1, 2} ∩ {2, 3} = {2}.
Some basic properties of intersections:
A ∩ B = B ∩ A.A ∩ (B ∩ C) = (A ∩ B) ∩ C.A ∩ B ⊆ A.A ∩ A = A.A ∩ U = A.A ∩ ∅ = ∅.A ⊆ B if and only if A ∩ B = A.ComplementsEdit
The relative complement
of B in A
The complement of A in U
The symmetric difference of A and B
Main article: Complement (set theory)
Two sets can also be "subtracted". The relative complement of B in A (also called the set-theoretic difference of A and B), denoted by A \ B (or A − B), is the set of all elements that are members of A but not members of B. Note that it is valid to "subtract" members of a set that are not in the set, such as removing the element green from the set {1, 2, 3}; doing so has no effect.
In certain settings all sets under discussion are considered to be subsets of a given universal set U. In such cases, U \ A is called the absolute complement or simply complement of A, and is denoted by A′.
A′ = U \ A
Examples:
{1, 2} \ {1, 2} = ∅.{1, 2, 3, 4} \ {1, 3} = {2, 4}.If U is the set of integers, E is the set of even integers, and O is the set of odd integers, then U \ E = E′ = O.
Some basic properties of complements:
A \ B ≠ B \ A for A ≠ B.A ∪ A′ = U.A ∩ A′ = ∅.(A′)′ = A.∅ \ A = ∅.A \ ∅ = A.A \ A = ∅.A \ U = ∅.A \ A′ = A and A′ \ A = A′.U′ = ∅ and ∅′ = U.A \ B = A ∩ B′.if A ⊆ B then A \ B = ∅.
An extension of the complement is the symmetric difference, defined for sets A, B as
{\displaystyle A\,\Delta \,B=(A\setminus B)\cup (B\setminus A).}
For example, the symmetric difference of {7, 8, 9, 10} and {9, 10, 11, 12} is the set {7, 8, 11, 12}. The power set of any set becomes a Boolean ring with symmetric difference as the addition of the ring (with the empty set as neutral element) and intersection as the multiplication of the ring.
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In mathematics, a set is a collection of distinct objects, considered as an object in its own right. For example, the numbers 2, 4, and 6 are distinct objects when considered separately, but when they are considered collectively they form a single set of size three, written {2, 4, 6}. The concept of a set is one of the most fundamental in mathematics. Developed at the end of the 19th century, set theory is now a ubiquitous part of mathematics, and can be used as a foundation from which nearly all of mathematics can be derived. In mathematics education, elementary topics from set theory such as Venn diagrams are taught at a young age, while more advanced concepts are taught as part of a university degree.
There are several fundamental operations for constructing new sets from given sets.
UnionsEdit
The union of A and B, denoted A ∪ B
Main article: Union (set theory)
Two sets can be "added" together. The unionof A and B, denoted by A ∪ B, is the set of all things that are members of either A or B.
Examples:
{1, 2} ∪ {1, 2} = {1, 2}.{1, 2} ∪ {2, 3} = {1, 2, 3}.{1, 2, 3} ∪ {3, 4, 5} = {1, 2, 3, 4, 5}
Some basic properties of unions:
A ∪ B = B ∪ A.A ∪ (B ∪ C) = (A ∪ B) ∪ C.A ⊆ (A ∪ B).A ∪ A = A.A ∪ U = U.A ∪ ∅ = A.A ⊆ B if and only if A ∪ B = B.IntersectionsEdit
Main article: Intersection (set theory)
A new set can also be constructed by determining which members two sets have "in common". The intersection of A and B, denoted by A ∩ B, is the set of all things that are members of both A and B. If A ∩ B = ∅,then A and B are said to be disjoint.
The intersection of A and B, denoted A ∩ B.
Examples:
{1, 2} ∩ {1, 2} = {1, 2}.{1, 2} ∩ {2, 3} = {2}.
Some basic properties of intersections:
A ∩ B = B ∩ A.A ∩ (B ∩ C) = (A ∩ B) ∩ C.A ∩ B ⊆ A.A ∩ A = A.A ∩ U = A.A ∩ ∅ = ∅.A ⊆ B if and only if A ∩ B = A.ComplementsEdit
The relative complement
of B in A
The complement of A in U
The symmetric difference of A and B
Main article: Complement (set theory)
Two sets can also be "subtracted". The relative complement of B in A (also called the set-theoretic difference of A and B), denoted by A \ B (or A − B), is the set of all elements that are members of A but not members of B. Note that it is valid to "subtract" members of a set that are not in the set, such as removing the element green from the set {1, 2, 3}; doing so has no effect.
In certain settings all sets under discussion are considered to be subsets of a given universal set U. In such cases, U \ A is called the absolute complement or simply complement of A, and is denoted by A′.
A′ = U \ A
Examples:
{1, 2} \ {1, 2} = ∅.{1, 2, 3, 4} \ {1, 3} = {2, 4}.If U is the set of integers, E is the set of even integers, and O is the set of odd integers, then U \ E = E′ = O.
Some basic properties of complements:
A \ B ≠ B \ A for A ≠ B.A ∪ A′ = U.A ∩ A′ = ∅.(A′)′ = A.∅ \ A = ∅.A \ ∅ = A.A \ A = ∅.A \ U = ∅.A \ A′ = A and A′ \ A = A′.U′ = ∅ and ∅′ = U.A \ B = A ∩ B′.if A ⊆ B then A \ B = ∅.
An extension of the complement is the symmetric difference, defined for sets A, B as
{\displaystyle A\,\Delta \,B=(A\setminus B)\cup (B\setminus A).}
For example, the symmetric difference of {7, 8, 9, 10} and {9, 10, 11, 12} is the set {7, 8, 11, 12}. The power set of any set becomes a Boolean ring with symmetric difference as the addition of the ring (with the empty set as neutral element) and intersection as the multiplication of the ring.
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