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Example

If A = {2, 4, 8, 12 }, B = {3, 4, 5, 8 } and U = { x | x Î N and x < 13 },

find A¢ ; B¢ , (A Ç B)¢ and verify (A Ç B)¢ = A¢ È B¢

Solution :

A¢ = U - A = { 1, 3, 5, 6, 7, 9, 10, 11 }

B ¢ = U - B = { 1, 2, 6, 7, 9, 10, 11, 12 }

Now A Ç B = { 4, 8 }

\ ( A Ç B)¢= U - ( A Ç B )={1, 2, 3, 5, 6, 7, 9, 10, 11, 12}

Also A¢ U B¢ = { 1, 2, 3, 5, 6, 7, 9, 10, 11, 12 } . . . (II)

\ ( A Ç B)¢ = A¢ È B¢ . . . (from (I) and (II) )

Example

If U = { x | x £ 8, x Î W},

A = { x | 2x -1 £ 8 ; x Î W},

B = { x | 5x - 1 £ 14 ; x Î N }

Find (1) A and B (2) (A È B)¢ (3) A¢ Ç B¢

(4) Hence show that ( A È B )¢ = A¢ Ç B¢

Solution :

(1) A = { x | 2x -1 £ 8; x Î W}

     Now 2x -1 £ 8

     \ 2x - 1 £ 8

     \ x £ 9/2

     \ A = { x | x £ 9/2 , x Î W} = { 0, 1, 2, 3, 4 }

     and B = { x | 5x - 1 £ 14, x Î N }

     Now 5x - 1 £ 14

      \ 5x £ 15

     \ x £ 3

     \ B = { x | x £ 3, x Î N } = { 1, 2, 3 }

(2) A È B = { 0, 1, 2, 3, 4 } È { 1, 2, 3 }={0, 1, 2, 3, 4 }

(3) A¢ = U - A = { 5, 6, 7, 8 } and

     B¢ = U - B = { 0, 4, 5, 6, 7, 8 }

     \ A¢ Ç B¢ = { 5, 6, 7, 8 } . . . (I)

     and (A È B)¢ = U - ( A È B ) = { 5, 6, 7, 8 } . . . (II)

(4) From (I) and (II) we get ( A È B )¢ = A¢ Ç B¢



Example

Given U = { x | x £ 10 , x ÎN },

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

Prove that (a) (A È B ) Ç B¢ = A if and only if A Ç B = f

Solution :

(a) A È B = { 2, 4, 6 }È{ 1, 3, 5, 7 }= { 1, 2, 3, 4, 5, 6, 7 }

and B¢ = U - B = { 2, 4, 6, 8, 9, 10 }

Therefore, ( A È B) Ç B¢ = { 1, 2, 3, 4, 5, 6, 7 } Ç { 2, 4, 6, 8, 9, 10 } = { 2, 4, 6 } = A

Index

2.1 Sets
2.2 Operations on Sets
2.3 The Algebra of Sets

Chapter 3





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