1. A = {1,2,3,4}, B = {3,4,5}, C = {1,3,5}, Prove that A intersect (B union C) = (A intersect B) union (A intersect C) ...

Solution:
Here `A={1,2,3,4}`,`B={3,4,5}`,`C={1,3,5}`

To find LHS = `A ∩ (B uu C)`

`B uu C = {3,4,5} uu {1,3,5}`

`= {color{red}{3},4,color{green}{5}} uu {1,color{red}{3},color{green}{5}}`

`= {3,4,5,1}`

`A nn (B uu C) = {1,2,3,4} nn {3,4,5,1}`

`= {color{red}{1},2,color{green}{3},color{blue}{4}} nn {color{green}{3},color{blue}{4},5,color{red}{1}}`

`= {1,3,4}`

`:. A ∩ (B uu C) = {1,3,4} ->(1)`

To find RHS = `(A ∩ B) uu (A ∩ C)`

`A nn B = {1,2,3,4} nn {3,4,5}`

`= {1,2,color{red}{3},color{green}{4}} nn {color{red}{3},color{green}{4},5}`

`= {3,4}`

`A nn C = {1,2,3,4} nn {1,3,5}`

`= {color{red}{1},2,color{green}{3},4} nn {color{red}{1},color{green}{3},5}`

`= {1,3}`

`(A ∩ B) uu (A ∩ C) = {3,4} uu {1,3}`

`= {color{red}{3},4} uu {1,color{red}{3}}`

`= {3,4,1}`

`:. (A ∩ B) uu (A ∩ C) = {3,4,1} ->(2)`

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

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

Solution:
`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}`

`C={x<=3; x in N}`
`:.C={1,2,3}`

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

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

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

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

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

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

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

`:. A uu (B uu C) = {1,2,3,4,5,6,7,8} ->(1)`

To find RHS = `(A uu B) uu C`

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

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

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

`:. (A uu B) uu C = {1,2,3,4,5,6,7,8} ->(2)`

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

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