Question

Find the intersection of each pair of sets of question 1 above.
If A={3,5,7,9,11 }, B = {7,9, 11, 13), C = {11, 13, 15}and D= {15, 17): find
(i) AB
(ii) Boc
(ii) An CD
(iv) Anc
(v) BD
(vi) An (BUC)
(vii) AD (viii) An (BUD) (ix) (AB)(BUC)
(x) (AOD (BUC)​

Answer 1

Given:

A = {3, 5, 7, 9, 11 }, B = {7, 9, 11, 13}, C = {11, 13, 15} and D = {15, 17}

We have to find:

i) A ∩ B      ii) B ∩ C                 iii) A ∩ C ∩ D    iv) A ∩ C

v) B ∩ D    vi)  A ∩ (B ∪ C)      vii) A ∩ D           viii) A ∩ (B ∪ D)

ix) (A ∩ B) ∩ (B ∪ C)                x) (A ∪ D) ∩ (B ∪ C)

Solution:

i) A ∩ B

= {3, 5, 7, 9, 11 } ∩ {7, 9, 11, 13}

= {7, 9, 11}

ii) B ∩ C

= {7, 9, 11, 13} ∩ {11, 13, 15}

= {11, 13}

iii) A ∩ C ∩ D

= (A ∩ C) ∩ D

= ({3, 5, 7, 9, 11 } ∩ {11, 13, 15}) ∩ {15, 17}

= {11} ∩ {15, 17}

= { } = Ф

iv) A ∩ C

= {3, 5, 7, 9, 11 } ∩ {11, 13, 15}

= {11}

v) B ∩ D

=  {7, 9, 11, 13} ∩ {15, 17}

= { } = Ф

vi)  A ∩ (B ∪ C)

= {3, 5, 7, 9, 11} ∩ ({7,9, 11, 13} ∪ {11, 13, 15})

= {3, 5, 7, 9, 11} ∩ {7, 9, 11, 13, 15}

= {7, 9, 11}

vii) A ∩ D

= {3, 5, 7, 9, 11} ∩ {15, 17}

= { } = Ф

viii) A ∩ (B ∪ D)

= {3, 5, 7, 9, 11} ∩ ({7, 9, 11, 13} ∪ {15, 17})

= {3, 5, 7, 9, 11} ∩ {7, 9, 11, 13, 15,17}

= {7, 9, 11}

ix) (A ∩ B) ∩ (B ∪ C)

= ({3, 5, 7, 9, 11} ∩ {7, 9, 11, 13}) ∩ ({7, 9, 11, 13} ∪ {11, 13, 15})

= {7, 9, 11} ∩ {7, 9, 11, 13, 15}

= {7, 9, 11}

x) (A ∪ D) ∩ (B ∪ C)

= ({3, 5, 7, 9, 11} ∪ { 15, 17}) ∩ ({7, 9, 11, 13} ∪ {11, 13, 15})

= {3, 5, 7, 9, 11, 15, 17} ∩ {7, 9, 11, 15}

= {7, 9, 11, 15}

Answer 2

Given:

A = {3, 5, 7, 9, 11 }

B = {7, 9, 11, 13}

C = {11, 13, 15}

D = {15, 17}

To find:

i) A ∩ B      

ii) B ∩ C                

iii) A ∩ C ∩ D    

iv) A ∩ C

v) B ∩ D    

vi)  A ∩ (B ∪ C)    

vii) A ∩ D          

viii) A ∩ (B ∪ D)

ix) (A ∩ B) ∩ (B ∪ C)                

x) (A ∪ D) ∩ (B ∪ C)

Solution:

i) A ∩ B

 {3, 5, 7, 9, 11 } ∩ {7, 9, 11, 13}

Common Elements are 7, 9, 11

A ∩ B {7, 9, 11}

ii) B ∩ C

 {7, 9, 11, 13} ∩ {11, 13, 15}

Common Elements are 11 , 13

B ∩ C = {11, 13}

iii) A ∩ C ∩ D

({3, 5, 7, 9, 11 } ∩ {11, 13, 15}) ∩ {15, 17}

No element is common in all 3

A ∩ C ∩ D = null set =  { } = Ф

iv) A ∩ C

 {3, 5, 7, 9, 11 } ∩ {11, 13, 15}

Common Element is 11 only

A ∩ C = {11}

v) B ∩ D

  {7, 9, 11, 13} ∩ {15, 17}

No element is common

B ∩ D = null set =  { } = Ф

vi)  A ∩ (B ∪ C)

 {3, 5, 7, 9, 11} ∩ ({7,9, 11, 13} ∪ {11, 13, 15})

 {3, 5, 7, 9, 11} ∩ {7, 9, 11, 13, 15}

A ∩ (B ∪ C) = {7, 9, 11}

vii) A ∩ D

= {3, 5, 7, 9, 11} ∩ {15, 17}

No element is common

A ∩ D = null set =  { } = Ф

viii) A ∩ (B ∪ D)

= {3, 5, 7, 9, 11} ∩ ({7, 9, 11, 13} ∪ {15, 17})

= {3, 5, 7, 9, 11} ∩ {7, 9, 11, 13, 15,17}

A ∩ (B ∪ D) = {7, 9, 11}

ix) (A ∩ B) ∩ (B ∪ C)

= ({3, 5, 7, 9, 11} ∩ {7, 9, 11, 13}) ∩ ({7, 9, 11, 13} ∪ {11, 13, 15})

= {7, 9, 11} ∩ {7, 9, 11, 13, 15}

= {7, 9, 11}

(A ∩ B) ∩ (B ∪ C) = {7, 9, 11}

x) (A ∪ D) ∩ (B ∪ C)

= ({3, 5, 7, 9, 11} ∪ { 15, 17}) ∩ ({7, 9, 11, 13} ∪ {11, 13, 15})

= {3, 5, 7, 9, 11, 15, 17} ∩ {7, 9, 11, 15}

= {7, 9, 11, 15}

(A ∪ D) ∩ (B ∪ C) = {7, 9, 11, 15}

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