SET

SET:


A set is a collection of definite well defined objects.
A set is a collection of objects which are distinct from each other.


A set is usually denoted by capital letter, i.e, A, B, S, T, G etc.
A set elements are denoted by small letter, i.e, a, b, s, t etc.


CONSTRUCTION OF SET:


In construction of set, two methods are commonly used-


1) Roster Method (Enumeration)- In this method we prepare a list of objects forming the set, writing the elements one after another between a pair of curly brackets.
For example:
A = {a, b, c, d}.


2) Description Method- In this method we describe the set in symbolic language.
For example:
A set of integer numbers which is divisible by 3 is written as,
A = {x : x is an integer divisible by 3}




TYPES OF SET:


1) Singleton set- If a set consisting only 1 element is known as singleton set.
For example:
A = {a}.


2) Finite set- If a set consisting finite number of elements is known as finite set.
For example:
A = {2, 4, 6, 8}.


3) Infinite set- If a set consisting infinite number of elements is known as infinite set.
For example-
The set of all natural numbers.
A = {1, 2, 3,......}


4) Equal sets- Two sets A and B consisting of the same elements is known as equal set.
For example:
A = {a, b, c, d} and
B = {a, b, c, d}


5) Empty set: If a set consisting no elements is known as empty set or null set or void set.
For example:
A = { }


6) Subset- Suppose A is a given set, and any set B exist exist whose elements are also an element of A,than B is called subset of A.
For example:
A = {1, 2, 3, 4, 5, 6, 7, 8} and B = {2, 4, 6, 8}
Than, B ⊆ A. (read as B is the subset of A)


Now take another example;
A = {1, 2, 3, 4, 5, 6, 7, 8} and B = {1, 2, 3, 4, 5, 6, 7, 8}
Than, B ⊆ A. (read as B is the subset of A)


7) Proper Subset- If B is the subset of A, and B≠A, then B is proper subset of A.
For example:
A = {1, 2, 3, 4, 5, 6, 7, 8} and B = {2, 4, 6, 8}
Than, B ⊂ A. (read as B is the proper subset of A)


8) Power set- The set of all subset of a set A, is known as power set of A.
For example:
A = {a, b, c}
Than
Power set,P(A) = {{∅ }, {a}, {b}, {c}, {d}, {ab}, {ac}, {ad}, {bc}, {bd}, {cd}, {abc}}





Problems based on above study:


Prob. If A, B and C are three sets, then prove that-
(A-B) ∩ (A-C) = A-(B∪C)
Solution: Click here


Prob. If  A and B are two sets, then prove that-
(A-B) ∪ (B-A) = (A ∪ B) - (A ∩ B)
Solution: Click here

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