Sets in Mathematics
A set is a collection of distinct objects, considered as an object in its own right.
For example, a set of vowels in the English alphabet can be written as:
A = {a,e,i,o,u}
Universal Set:
The universal set (denoted as U) is the set that contains all the objects under consideration, usually within a particular context.
Example: If we are talking about the English alphabet, the universal set could be:
U = {a,b,c,....,z}
Finite Set:
A finite set has a limited number of elements.
Example: The set of prime numbers less than 10:
P = {2,3,5,7}
Infinite Set:
An infinite set has an unlimited number of elements.
Example: The set of all natural numbers:
N = {1,2,3,4,...}
Empty Set (Null Set):
An empty set has no elements and is denoted by ∅ or {}.
Example: The set of all natural numbers less than 1:
∅ = {}
Subset:
A set A is a subset of set B if all elements of A are also elements of B, denoted as A⊆B.
Example: A = {1,2} is a subset of B = {1,2,3}.
Equal Set:
Two sets A and B are equal if they have exactly the same elements, denoted as A = B.
Example: A = {1,2,3} and B = {3,2,1} are equal sets.
Proper Subset:
A set A is a proper subset of set B if A is a subset of B and A≠B, denoted as A⊂B.
Example: A = {1,2} is a proper subset of B = {1,2,3}.
Set of Sets:
A set of sets is a set whose elements are themselves sets.
Example: S = {{1,2},{3,4}}
Power Set:
The power set of a set A is the set of all subsets of A, including the empty set and A itself, denoted as P(A).
Example: If A = {1,2}, then: P(A) = {∅,{1},{2},{1,2}}
Operation of Sets
Operations on sets involve combining, comparing, or modifying sets using specific rules. Common operations include union, intersection, difference, and complement.
Union of Sets
The union of two sets A and B (denoted as A ∪ B) is the set of all elements that are in A, in B, or in both.
Example:
A = {1,2,3}
B = {3,4,5}
A ∪ B = {1,2,3,4,5}
Intersection of Sets
The intersection of two sets A and B (denoted as A ∩ B) is the set of all elements that are both in A and B.
Example:
A = {1,2,3}
B = {3,4,5}
A ∩ B = {3}
Difference of Two Sets
The difference of two sets A and B (denoted as A−B or A∖B) is the set of all elements that are in A but not in B.
Example:
A = {1,2,3}
B = {3,4,5}
A - B = {1,2}
Complement of a Set
The complement of a set A (denoted as A′ or Ac) is the set of all elements in the universal set U that are not in A.
Example:
If the universal set U = {1,2,3,4,5} and A = {1,2} then:
A′ = {3,4,5}
Disjoint Set
Two sets A and B are disjoint if they have no elements in common, i.e., A ∩ B = ∅
Example:
A = {1,2,3}
B = {4,5,6}
A ∩ B = ∅
So, A and B are disjoint sets.
Venn Diagram
A Venn diagram is a visual representation of sets and their relationships using circles.
Example:
If A = {1,2,3} and B = {3,4,5}, a Venn diagram would show two overlapping circles, one for each set, with the overlap representing {3}.
Cartesian Product of Sets
The Cartesian product of two sets A and B (denoted as A×B) is the set of all ordered pairs (a,b) where a ∈ A and b ∈ B.
Example:
A = {1,2}
B = {x,y}
A × B = {(1,x),(1,y),(2,x),(2,y)}
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