UNION AND PROPERTIES OF UNION

DEFINITION OF UNION

The set of all elements of SET A and SET B is called the union of sets

where repeating element eliminates symbolically we denote the sign " U 

union of two set A and B is denoted by the sign AUB says set of all element A and B Words union mean together and we read AUB A union B

AUB = { set of all elements, such that, all elements of A and B } mathematically we write

AUB = { y: y is part of A or part of B }

if we have set A = { 1,2,3,4,5,6,7 } and B = { 5,6,7,8,9 }

AUB ={ 1,2,3,4,5,6,7 } U { 5,6,7,8,9 }
AUB = { 1,2,3,4,5,6,7,8,9 }

A U B diagram is shown as

we take an example to more explanation
Example 1:

 if we have set S = { 15,16 ,17,20,25} T = { 12, 13,14 }

 S U T = { 12,13,14,15,16,17,20,25}

Example 2
 R = { 0,1,7,9,11.17 } and S = { 5,7,,11,15 }

  R U S = { 0.1,7,9,11,15,17 }         we cannot repeat element in union of set
EXAMPLE 3

  if we have P set and empty set then the union of sets
 P = { 6,7,8,9,10,11 }         and             Q = { φ }

 P U Q = { 6,7,8,9,10,11 } U { φ }

 PUQ = { 6,7,8,9,10,11 }
UNION OF THREE SET

first, we take the union of two set answers with union three

👰👰
  If we have set B = { 1,2,8 } C = { 6, 7,9 } D = { 3,5,8 }

 ( BUC ) U D = { ( 1,2,8 ) U ( 6,7,8) } U (3,5,8 )}

 ( BUC ) U D = {( 1 , 2 ,6 , 7 , 8 ) U ( 3, 5 , 8 }

 ( BUC ) U D = { 1 ,2 ,5 , 6 ,7 , 8 }

CONCLUSION ;
 1; A ⊆ AUB and
 2: B ⊆ AUB
 3 ; AUB = BUA

 we write these forms when sets in ROOSTER FORM

PROPERTIES OF UNION OF SET
 1

 AUB = BUA                    ( COMMUTATIVE LAW )

  2 : 

 A U (B U C ) = ( AUB) UC         ( ASSOCIATIVE LAW)

 3 :

 A U φ = A                   ( UNION LAW OF IDENTITY )

  4 :

 BUB = B and AUA =A         (LAW OF IDOMPOTANT)

  5 :    

  U U B = B                             ( UNIVERSAL SET )

6: 

AU φ = φ U A        ( empty set with union of any

 SET A, B, C himself SET )

WHAT ARE THE PROPERTY OF "" REAL NUMBER ""
 (1) prove that AUB = BUA  (by commutative law)

  set A = { 11,12,13} and B = { 7,8,9}

 L.H.S = AUB
 = { 11,12,13} U { 7,8,9 }
 = { 7,8,9,11,12,13 }

  R.H.S = BUA

 = { 7,8,9,} U { 11,12,13 }
 = {7,8,9,11,12,13 }

 HENCE              AUB = BUA

(2) PROVE THAT ASSOCIATIVE LAW OF UNION

 if we have set A = { 11,12,13} and B = { 7,8,9} and C = { 0,1 }

 A U {B U C} = { AUB } U C

 L.H.S = A U {B U C}

 ={ 11,12,13} U [{ ( 7,8,9 } U { 0,1 }]
= { 11,12,13} U { 0,1,7,8,9 }
 = { 0,1,7,8,9,11,12,13 }

 R.H.S= { AUB }UC

= [ { 11,12,13}U { 7,8,9} ] U { 0,1 }
 = { 7,8,9,11,12,13 } U { 0,1}
 = { 0,1,7,8,9,11,12,13 }

 SO        L.H.S = R . H .S