if U=(1,2,3,4,5,6,7,8,9,10)andA=(1,5,7)and B=(3,5,9,10) then Find the value of (AunionB)'n(AnB)'
Answers
Answer :
(A U B)' ∩ (A ∩ B)' = { 2 , 4 , 6 , 8 }
Note :
★ Set : A well defined collection of distinct objects is called a set .
★ Union of two sets : The union of two sets A and B is the set of all those elements which are either in A or in B or in both .
→ This set is denoted by A U B .
★ Intersection of two sets : The intersection of two sets A and B is the set of all those elements which are in common in both A and B .
→ This set is denoted by A ∩ B .
Solution :
→ Given :
• U = { 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 }
• A = { 1 , 5 , 7 }
• B = { 3 , 5 , 9 , 10 }
→ To find :
• (A U B)' ∩ (A ∩ B)'
Firstly ,
Let's find A U B ,
=> A U B = { 1 , 5 , 7 } U { 3 , 5 , 9 , 10 }
=> A U B = { 1 , 3 , 5 , 7 , 9 , 10 }
Thus ,
=> (A U B)'
= U - (A U B)
= { 1,2,3,4,5,6,7,8,9,10 } - { 1,3,5,7,9,10 }
= { 2 , 4 , 6 , 8 }
Secondly ,
Let's find A ∩ B ,
=> A ∩ B = { 1 , 5 , 7 } ∩ { 3 , 5 , 9 , 10 }
=> A ∩ B = { 5 }
Thus ,
=> (A ∩ B)' = U - (A ∩ B)
=> (A ∩ B)' = { 1,2,3,4,5,6,7,8,9,10 } - { 5 }
=> (A ∩ B)' = { 1 , 2 , 3 , 4 , 6 , 7 , 8 , 9 , 10 }
Finally ,
=> (A U B)' ∩ (A ∩ B)'
= { 2,4,6,8 } ∩ { 1 , 2 , 3 , 4 , 6 , 7 , 8 , 9 , 10 }
= { 2 , 4 , 6 , 8 }
Hence,
(A U B)' ∩ (A ∩ B)' = { 2 , 4 , 6 , 8 }