Answers

2016-06-20T19:04:15+08:00

A set that has no element should be called as Empty set. Another name for Empty set could be Null set and Void set. Number of element in set X is represented as n(X). The empty set is denoted as Φ. Thus, n(Φ) = 0. The cardinality of an empty set is zero since it has no element.

Singleton Set
A set that has one and only one element should be called as Singleton set. Sometimes, it is known as unit set. The cardinality of singleton is one. If A is a singleton, then we can express it as
A = {x : x = A}

Example: Set A = {5} is a singleton set.Finite and Infinite Set
A set that has predetermined number of elements or finite number of elements are said to be Finite set. Like {1 ,2, 3, 4, 5, 6} is a finite set whose cardinality is 6, since it has 6 elements. 

Otherwise, it is called as infinite set. It may be uncountable or countable. The union of some infinite sets are infinite and the power set of any infinite set is infinite.

Examples: 
Set of all the days in a week is a finite set.Set of all integers is infinite set.Union of Sets
Union of two or else most numbers of sets could be the set of all elements that belongs to every element of all sets. In the union set of two sets, every element is written only once even if they belong to both the sets. This is denoted as ‘∪’. If we have sets A and B, then the union of these two is A U B and called as A union B.

Mathematically, we can denote it as A U B = {x : x ∈∈ A or x∈∈ B}

The union of two sets is always commutative i.e.A U B = B U A.

Example: A = {1,2,3}

B = {1,4,5}

∪∪ B = {1,2,3,4,5}Intersection of Sets
It should be the set of elements that are common in both the sets. Intersection is similar to grouping up the common elements. The symbol should be denoted as ‘∩’. If A and B are two sets, then the intersection is denoted as A ∩∩ B and called as A intersection B and mathematically, we can write it as
A∩B={x:x∈A∧x∈B}A∩B={x:x∈A∧x∈B}

Example: A = {1,2,3,4,5}
B = {2,3,7}
∩∩ B = {2,3}Difference of Sets
The difference of set A to B should be denoted as A - B. That is, the set of element that are in set A not in set B is
A - B = {x: x ∈∈ A and x ∉ B}

And, B - A is the set of all elements of the set B which are in B but not in A i.e.
B - A = {x: x ∈∈ B and x ∉ A}.

Example: 

If A = {1,2,3,4,5} and B = {2,4,6,7,8}, then
A - B = {1,3,5} and B - A = {6,7,8}Subset of a Set
In set theory, a set P is the subset of any set Q, if the set P is contained in set Q. It means, all the elements of the set P also belongs to the set Q. It is represented as '⊆’ or P ⊆⊆ Q.

Example:

A = {1,2,3,4,5}
B = {1,2,3,4,5,7,8}
Here, A is said to be the subset of B.Disjoint Sets
If two sets A and B should have no common elements or we can say that the intersection of any two sets A and B is the empty set, then these sets are known as disjoint sets i.e. A ∩∩ B = ϕϕ. That means, when this condition n (A ∩ B) = 0 is true, then the sets are disjoint sets.

Example: 
A = {1,2,3}
B = {4,5}
n (A ∩ B) = 0.
Therefore, these sets A and B are disjoint sets.Equality of Two SetsBack to Top
Two sets are said to be equal or identical to each other, if they contain the same elements. When the sets P and Q is said to be equal, if P ⊆ Q and Q ⊆ P, then we will write as P = Q.

Examples:
 
If A = {1,2,3} and B = {1,2,3}, then A = B.Let P = {a, e, i, o, u} and B = {a, e, i, o, u, v}, then P ≠≠ Q, since set Q has element v as the extra element.

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