SETS
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When working with sets, a reference superset that contains all the sets and their subsets in context (i.e. the sets which are currently under consideration) is called a universal set. It provides a base set from which all of the sets of interest can be derived or deduced. The universal set concept is needed at times in solving computational problems related to sets and providing certain axioms and in the application of Venn diagrams. When dealing with multiple collections of the same objects, then a set that contains all the possible objects and all sets and subsets formed from these elements is called a superset of every other set or universal set. It is generally denoted by the U. There are no laws […] READ MORE...

Partitioning of a set is distributing the member elements of a set among a group of non-empty subsets in such a way that each member lies in only one of these subsets. ⇒ ∅ ( Empty Set ) cannot be the partition of any set. Examples ⇒ The set { 1, 2, 3 } can be partitioned in the below subsets :{ 1 } , { 2, 3 }{ 2 } , { 1, 3 }{ 3 } , { 1, 2 }{1}, {2}, {3} READ MORE...

Two sets A and B are called disjoint when they have no element in common (except the empty set ∅ }. READ MORE...

The set of all possible subsets of a set S is called the power set of S, written as P(S). Examples ⇒ The power set of { ‘a’ } is { ∅, {‘a’} }The power set of { 2, 3 } is { ∅, {2}, {3}, {2,3} }The power set of {1, 2, 3 } is { ∅, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3} } If there are n members in a set S i.e. cardinality of set |S| = n, then there are 2n subsets possible.Hence the power set would contain 2n elements. Or we can say that the cardinality of the power set is |P| is 2n. READ MORE...

If A is a subset of B but A ≠ B, then A is called the proper subset of B, and B is called the proper superset of A. This relationship is represented as below A ⊂ B ( A is a proper subset of A )B ⊃ A ( B is a proper superset of A ) Examples ⇒ { x2 | x ∈ N } ⊂ N ( not all natural numbers are squares ){ 1, 2, 3 } ⊂ { 1, 2, 3, 4 }{ 1, 2, 3, 3 } ⊂ { 1, 2, 3, 4 } READ MORE...

If there are two sets A & B such that every element of A is also in B, then A is called a subset of B. In other words, A is contained in B.  B is called the superset of A. In the set theory, this relationship is depicted as below A ⊆ B ( A is a subset of B) B ⊇ A ( B is a superset of A ) ⇒ ∅ (empty set) is the subset of every set ⇒ A set S is a subset of itself. Examples ⇒ { 1 } ⊆ { 1, 2, 3 } ∅ ⊆ { 1, 2, 3 } { 1, 2, 3 } ⊆ { 1, 2, 3 } { x2 | x ∈ N […] READ MORE...

Two sets A and B are said to be equivalent(≡) if each element of A is also an element of B and each element of B is also an element of A. If elements are repetitive in one set, then it is not required for it to repeat in the other set for the two sets to be equivalent. Examples ⇒ { 1, 2, 3 } ≡ { 1, 3, 2} { 1, 2, 3 } ≡ { 1, 3, 2, 2, 1, 2, 3, 3 } { 1, 2, 3 } ≢ { 1, 3, 2, 2, 1, 2, 3, 3, 4 } Since 4 is not in first set READ MORE...