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Chapter 1
set
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A set is a collection of well-defined and distinct objects called elements of
the set
...
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Sets are represented by capital letters, and elements are represented by
small letters or symbols
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Example: C = {}
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Finite Set: A set that has a finite number of elements is called a finite set
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Example: E = {1, 2, 3, 4,
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Example: F = {5}
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Equal Sets: Two sets are equal if they have the same elements, regardless
of the order or repetition of the elements
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Subset: A set A is said to be a subset of set B if every element of A is also an
element of B
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Universal Set: The set of all elements under consideration in a given
situation is called the universal set, denoted by the symbol 'U'
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Operations on Sets:
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Union: The union of two sets A and B is the set of all elements that belong
to either A or B, denoted by A ∪ B
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Intersection: The intersection of two sets A and B is the set of all elements
that belong to both A and B, denoted by A ∩ B
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Complement: The complement of a set A with respect to a universal set U is
the set of all elements in U that are not in A, denoted by A'
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There are several properties of set operations in set theory
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Commutative Property: The order of sets in a union or intersection does
not affect the result
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Associative Property: The grouping of sets in a union or intersection does
not affect the result
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Distributive Property: The distribution of a set operation over another set
operation results in an equivalent expression
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Identity Property: The union of a set with the empty set is the set itself, and
the intersection of a set with the universal set is the set itself
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Complement Property: The complement of a set with respect to a universal
set U is unique
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De Morgan's Laws: The complement of a union or intersection of sets is
equivalent to the intersection or union of the complements of the sets,
respectively
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