1. Examples
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}`
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)
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)
This material is intended as a summary. Use your textbook for detail explanation.
Any bug, improvement, feedback then
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