1. A = {1,2,3,4}, B = {3,4,5}, Find A intersect B ...

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

`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}`

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2. A = {x<=5; x in N}, B = {2<=x<=8; x in N}, Find A intersect B ...

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}`

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

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

`= {2,3,4,5}`

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

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

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

`= {2,3}`

`A nn (B ∩ C) = {1,2,3,4,5} nn {2,3}`

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

`= {2,3}`

`:. A ∩ (B ∩ C) = {2,3} ->(1)`

To find RHS = `(A ∩ B) ∩ C`

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

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

`= {2,3,4,5}`

`(A ∩ B) nn C = {2,3,4,5} nn {1,2,3}`

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

`= {2,3}`

`:. (A ∩ B) ∩ C = {2,3} ->(2)`

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

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