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Set Theory

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.

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Basics operations in Set Theory

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There are six basics operations in set theory. The names are,

Union

In basic set theory, the union of sets A and B is represented by A?B. That is group the values of the set A and set B.
 Union of sets is the set defined as

A $\cup$ B = {a | a $\in$ A or a $\in$ B } 

Intersection

     The intersection of the sets A and B represented by A ? B. It contains common elements of the set A and set B.
Intersection of sets is the set defined as

A $\cap$ B = {a | a $\in$ A and a $\in$ B }

Complement

Complement of set A is represented by Ac, is contain the all elements of universal set that are not members of A.

Difference

The difference of sets is denoted by A - B. If we subtract set B from set A, then A - B is the set of all element in A, but not in B.   
Difference of sets is the set defined as

A - B = { a | a $\in$ A and a $\notin$ B }  

Cartesian Product

In basics set theory, the set A and set B represented by a A x B.
Difference of sets is represented by

A x B = {(a, b) | a $\in$ A and b $\in$ B} 

Powers Set

 Power set is the set whose elements are all possible subsets of given set.

Laws in Set Theory

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Idempotent Laws

  • A?A = A
  • A?A = A

Commutative Laws

  • A?B = B?A
  • A?B = B?A

Associative Laws

(A?B) ? C = A?(B?C)

(A?B) ?C = A?(B?C)

Distributive Laws

  • A?(B?C) = (A?B) ?(A?C)
  • A?(B?C) = (A?B) ? (A?C)

Identity Laws

  • A?Ø = A
  • A?U = U
  • A?Ø = Ø
  • A?U = A

Complement Laws

  • A?A’ = U
  • (A’)’ = A
  • A?A’ = Ø
  • A - B = A?B’

DeMorgan’s Laws

  • (A?B)’ = A’ ? B’
  • (A?B)’ = A’ ? B’

Consistency Principle

  • A`sube` B iff A?B = B.
  • A`sube` B iff A?B = A.

Example

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Prove that 

    i) A?(B?C) = (A?B) ?(A?C)

   ii) A?(B?C) = (A?B) ? (A?C)

If A = {1, 3, 4, 5, 6}, B = {2, 3, 4, 5, 6} and C = {1, 6, 7, 8, 9}

Solution

     The given sets are A = {1, 3, 4, 5, 6}, B = {2, 3, 4, 5, 6} and C = {1, 6, 7, 8, 9}

i) A?(B?C) = (A?B) ?(A?C)

First solve left hand side, A?(B?C)

B?C = {6}

  A?(B?C) = { 1, 3, 4, 5, 6 }

The right side condition solve by the following way.

(A?B) ?(A?C)

     A?B = {1, 2, 3, 4, 5, 6}

     A?C = {1, 3, 4, 5, 6, 7, 8, 9}

     (A?B) ? (A?C) = {1, 3, 4, 5, 6}

So A?(B?C) = (A?B) ? (A?C).

ii) A ? (B?C) = (A?B) ? (A?C)

Take left hand side condition

A ? (B?C)

     B?C = {1, 2, 3, 4, 5, 6, 7, 8, 9}

     A ? (B?C) = {1, 3, 4, 5, 6}

Now take the right hand side condition .

(A?B) ? (A?C)

     A?B = {3, 4, 5, 6}

     A?C = {1, 6}

     (A?B) ? (A?C) = {1, 3, 4, 5, 6}

So A ? (B?C) = (A?B) ? (A?C).

Hence proved.


More topics in  Set Theory
Set Operations Laws of Set Theory
Cardinal Numbers How Many Elements Are in the Power Set?
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