Table of contents  
Definition  Set theory  
Types of Set  
Solved Examples for You 
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The set is a welldefined collection of definite objects of perception or thought.
If a set contains of only one element it is called to be a singleton set.
Examples:
As the name implies, a set with a finite or exact countable number of elements is called a finite set. If the set is nonempty, it is called a nonempty finite set.
Examples:
A set with an infinite number of elements is called an infinite set. In other words, if a given set is not finite, then it will be an infinite set.
Examples:
Finite and Infinite Sets
Two sets X and Y are said to be equal if they have exactly the same elements (irrespective of the order of appearance in the set). Equal sets are represented as X = Y. Otherwise, the sets are referred to as unequal sets, which are represented as X ≠ Y.
Any Set that does not contain any element is called the empty or null or void set. The symbol used to represent an empty set is – {} or φ (called phi).
Examples:
A set S is said to be a subset of set T if the elements of set S belong to set T, or you can say each element of set S is present in set T. Subset of a set is denoted by the symbol (⊂) and written as S ⊂ T and T is the superset of set S(T ⊇ S).
Example:
Example: A = {1,2,3} B = {1,2,3,4,5,6}
A ⊆ B, since all the elements in set A are present in set B.
B ⊇ A denotes that set B is the superset of set A.
In the given Venn diagram, A is the subset of set B.
Equivalent sets are those which have an equal number of elements irrespective of what the elements are .
Examples:
Two sets are disjoint sets if there are no common elements in both sets.
Example:
A = {1,2,3,4} B = {7,8,9,10}. Here, set A and set B are disjoint sets.
Power set of a set is defined as a set of every possible subset. It is denoted by P(A).
If the cardinality of A is n then the Cardinality of the power set is 2^{n} as every element has two options either to belong to a subset or not.
Example:
Any set which is a superset of all the sets under consideration is said to be universal set and is generally denoted U.
The universal set is represented by the letter U.
Example:
Let A = {1, 2, 3}
C = { 0, 1} then we can take
S = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} as universal set.
Question 1: If A = {x: x is an even natural number} and B = {y: y is the outcome of a dice roll}, determine the nature of the two sets.
Answer : A = {2, 4, 6, 8, 10, 12, 14, …} And B = {1, 2, 3, 4, 5, 6}. So, set A is an infinite set while set B is a finite set.
Question 2: If X = {1, 2, 3, 4, 5}, Y = {a, e, i, o, u} and Z = {u, o, a, i, e}; determine the nature of sets.
Answer: Since the pairs of sets X – Y, Y – Z as well as Z – X have the same number of elements, i.e. 5 they are EQUIVALENT sets. And sets Y and Z are also EQUAL sets because apart from having the number of elements the same, they also have the same elements, i.e. the alphabets of English vowel letters.
Question 3: What is the classification of sets in mathematics?
Answer: There are various kinds of sets like – finite and infinite sets, equal and equivalent sets, a null set. Further, there are a subset, power set, universal set in addition to the disjoint sets with the help of examples.
Question 4: What are the properties of sets?
Answer: The fundamental properties are that a set can consist of elements and that two sets are equal, if and only if every element of each set is an element of the other; this property is referred to as the extensionality of sets.
Question 5: If A = {R,O,Y,A,L} and B = {L,O,Y,A,L}, determine the type of the two sets to be equal or unequal.
Answer: A = {R,O,Y,A,L} and B = {L,O,Y,A,L}. So, sets A and B are unequal sets as the elements of set A are not similar to the elements of set B.
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