1. A = {2,3,6}, U = {1,2,3,4,5,6}, Find A' ...

Solution:
Here `U={1,2,3,4,5,6},A={2,3,6}`

`A' = {2,3,6}'`

`= {1,4,5}`

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2. A = {2,3,6}, U = {1,2,3,4,5,6}, Prove that (A')' = A ...

Solution:
Here `U={1,2,3,4,5,6},A={2,3,6}`

To find LHS = `A''`

`A' = {2,3,6}'`

`= {1,4,5}`

`A'' = {1,4,5}'`

`= {2,3,6}`

`:. A'' = {2,3,6} ->(1)`

To find RHS = `A`

`:. A = {2,3,6} ->(2)`

From (1) and (2)
`:. A'' = A` (proved)

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3. A = {x<=5; x in N}, B = {2<=x<=8; x in N}, U = {x<=10; x in N}, Prove that (A union B)' = A' intersect B' ...

Solution:
`U={x<=10; x in N}`
`:.U={1,2,3,4,5,6,7,8,9,10}`

`A={x<=5; x in N}`
`:.A={1,2,3,4,5}`

`B={2<=x<=8; x in N}`
`:.B={2,3,4,5,6,7,8}`

To find LHS = `(A uu B)'`

`A uu B = {1,2,3,4,5} uu {2,3,4,5,6,7,8}`

`= {1,color{red}{2},color{green}{3},color{blue}{4},color{maroon}{5}} uu {color{red}{2},color{green}{3},color{blue}{4},color{maroon}{5},6,7,8}`

`= {1,2,3,4,5,6,7,8}`

`(A uu B)' = {1,2,3,4,5,6,7,8}'`

`= {9,10}`

`:. (A uu B)' = {9,10} ->(1)`

To find RHS = `A' ∩ B'`

`A' = {1,2,3,4,5}'`

`= {6,7,8,9,10}`

`B' = {2,3,4,5,6,7,8}'`

`= {1,9,10}`

`A' nn B' = {6,7,8,9,10} nn {1,9,10}`

`= {6,7,8,color{red}{9},color{green}{10}} nn {1,color{red}{9},color{green}{10}}`

`= {9,10}`

`:. A' ∩ B' = {9,10} ->(2)`

From (1) and (2)
`:. (A uu B)' = A' ∩ B'` (proved)

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