subject: Set Theory Concepts [print this page] Set theory is the branch of mathematics that learned about the sets, which are the collections of objects. Even though any type of objects can be collected into a set, set theory is applied most often to objects that are related to mathematics. Here we will see about the concepts with examples of set theory.(Source:Wikipedia).
Basic Concepts in Set Theory
There are six Concepts available in set theory. The concepts are,
Union
The set A and B is symbolized by A?B .That is group the values of the set A and B.
Intersection
The set A and B symbolized by A nB. It means we only select the common values of the set A and B.
Complement
It is represented by Ac is the all values of U that are not components of A.
Difference
The sets A and B are the group of all objects. In that entity that is an element of accurately one A and B.
Cartesian Product
A x B is the Cartesian product of set A and B.
Powers Set
Whose elements are all possible subsets of A is called the power set of A.
The concept of a set is of fundamental importance in Mathematics. A football team is a set of players. A class is a set of students. The school library houses a set of books on Mathematics, a set of books on Physics, and so on.
Thus, we can say that a set is a well defined collection of objects. When we say "well-defined" it means that we must be given a rule or rules with the help of which we should be readily able to say whether a particular object is a member of the set or or not.
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Another example, vowels in the English alphabet form a set because any of the alphabet is either a vowel or a consonant. The collection of all honest people in a country is not a set, because the term "honest" is not well-defined.
If S is any set, every object in S is called an element of the set. For example, let S be the set of all natural numbers less than 100. Then, S = {1, 2, 3, 4, ......., 10, 11, ........97, 98, 99}
10 is an element of the set. Also, 36 is an element of the set. The fact that 10 is an element of the set expressed in symbols as 10 ? S which is read as "10 belongs to S" or "10 is an element of S".
Example
A={2,3,6,8,9,10} B={2,4,5,6,7,9} C={3,5,6,7,10,11}. Find the following conditions.
i)(A?B) ? C=A?(B?C)
ii)(AnB) nC=An(BnC)
Solution
The given sets are A={2,3,6,8,9,10} B={2,4,5,6,7,9} C={3,5,6,7,10,11}.
i)(A?B) ? C=A?(B?C)
Solve the left hand side condition.
(A?B) ? C
Here first find the A?B.
So we are joining the values of A and B.
A?B = {2,3, 4,5,6,7,8,9,10}
(A?B) ? C
Now joining the C set values also.
(A?B) ? C={2,3, 4,5,6,7,8,9,10, 11}
Solve the right side condition.
A?(B?C)
B?C = {2,3,4,5,6,7,910,11}
Now grouping the A set values with B?C.
A?(B?C)={ 2,3,4,5,6,7,8,9,10,11}
So (A?B) ? C=A?(B?C)
ii)(AnB) nC=An(BnC)
The left hand side is (AnB) nC. Solve this Condition.
