(i)  1,`1/2,1/3,........... 1/n.........`

(ii) `0, 1/3, 2/4,3/5,...........(n-1)/(n+1)............`

(iii) `1/2,1/4,1/8 ........ 1/2^n ...............`

(iv) `1,1/2,1/4,........... 1/2^(n-1) ...........`

(v) `1/2, - 2/3,3/4,...........(-1)^(n+1) n/(n+1)............`

(vi) `1/3, - 2/5,3/7............(-1)^(n-1) n/(2n+1)...........`

সমাধান:(ক) B \\ C
দেওয়া আছে,

`B = {1, 2, a}` এবং `C = {2, a, b}`

`:. ` B \\ C` = (1, 2, a} - {2, a, b}`

`= {1}`

`:.` B \\ C` = {1}` (Ans.)(খ)`A uu B`
দেওয়া আছে,

`A = {2, 3, 4}`এবং`B = {1, 2, a}`

`:.A uu B = {2, 3, 4} uu {1, 2, a}`

`= {1, 2, 3, 4, a}`

`:.A uu B = {1, 2, 3, 4, a}`  (Ans.)(গ)`A nn C`
দেওয়া আছে,
`A = {2, 3, 4}`এবং`C = {2, a, b}`

`:.A nn C = {2, 3, 4} nn {2, a, b}`

`= {2}`

`:.A nn C = {2}`     (Ans.)(ঘ)`A uu (B nn C)`
দেওয়া আছে,

`A = {2, 3, 4}, B = {1, 2, a}`

এবং `C = {2, a, b}`

এখানে,

`(B nn C) = {1, 2, a} nn {2, a, b}`

`= {2, a}`

`:.A uu (B nn C) = {2, 3, 4} uu {2, a}`

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

`:.A uu (B nn C) = {2, 3, 4, a}`  (Ans.)(ঙ)`A nn (B uu C)`
দেওয়া আছে,

`A = {2, 3, 4}, B = {1, 2, a}`

এবং `C = {2, a, b}`

এখানে,
`(B uu C) = {1, 2, a} uu {2, a, b}`

`= {1, 2, a, b}`

`:.A nn (B uu C) = {2, 3, 4} nn {1, 2, a, b}`

`= {2}`

`:.A nn (B uu C) = {2}`  (Ans.)

সমাধান:
(i). `(A uu B)' = A' nn B'`
সেট
উপাদান
`U`
`1, 2, 3, 4, 5, 6, 7`
`A`
`1, 3, 5`
`B`
`2, 4, 6`
`A uu B`
`1, 2, 3, 4, 5, 6`
`(A uu B)'`
`7`
`A'`
`2, 4, 6, 7`
`B'`
`1, 3, 5, 7`
`A' nn B'`
`7`
`:. (A uu B)' = A' nn B'` (সত্যতা যাচাই হলো)বিকল্প সমাধান:
দেওয়া আছে, 
`U = {1, 2, 3, 4, 5, 6, 7},`
`A = {1, 3, 5}` এবং `B = {2, 4, 6}`
এখানে, 
`A uu B = {1, 2, 5} uu {2, 4, 6}`
`= {1, 2, 3, 4, 5, 6}`
`:. (A uu B)' = U - (A uu B)`
`= {1, 2, 3, 4, 5, 6, 7} - {1, 2, 3, 4, 5, 6}`
`= {7}`
আবার,
`A' = U - A`
`= {1, 2, 3, 4, 5, 6, 7} - {1, 3, 5}`
`= {2, 4, 6, 7}`
`B' = U - B`
`= {1, 2, 3, 4, 5, 6, 7} - {2, 4, 6}`
`= {1, 3, 5, 7}`
`:. A' nn B' = {2, 4, 6, 7} nn {1, 3, 5, 7}`
`:. (A uu B)' = A' nn B'`  (সত্যতা যাচাই হলো)
(ii). `(B nn C)' = B' uu C'`
সেট
উপাদান
`U`
`1, 2, 3, 4, 5, 6, 7`
`B`
`2, 4, 6`
`C`
`3, 4, 5, 6, 7`
`B nn C`
`4, 6`
`(B nn C)'`
`1, 2, 3, 5, 7`
`B'`
`1, 3, 5, 7`
`C'`
`1, 2`
`B' uu C'`
`1, 2, 3, 5, 7`
`:. (B nn C)' = B' uu C'`   (সত্যতা যাচাই হলো)বিকল্প সমাধান:
দেওয়া আছে, `U = {1, 2, 3, 4, 5, 6, 7},`
`B = {2, 4, 6}` এবং `C = {3, 4, 5, 6, 7}`
এখানে,
`B nn C = {2, 4, 6} nn {3, 4, 5, 6, 7}`
`= {4, 6}`
`:. (B nn C)' = U - (B nn C)`
`= {1, 2, 3, 4, 5, 6, 7} - {4, 6}`
`= {1, 2, 3, 5, 7}`
আবার,
`B' = U - B`
`= {1, 2, 3, 4, 5, 6, 7} - {2, 4, 6}`
`= {1, 3, 5, 7}`
`C' = U - C`
`= {1, 2, 3, 4, 5, 6, 7} - {3, 4, 5, 6, 7}`
`= {1, 2}`
`:. B' uu C' = {1, 3, 5, 7} uu {1, 2}`
`= {1, 2, 3, 5, 7}`
`:. (B nn C)' = B' uu C'` (সত্যতা যাচাই হলো)
(iii). `(A uu B) nn C = (A nn C) uu (B nn C)`
সেট
উপাদান
`A`
`1, 3, 5`
`B`
`2, 4, 6`
`C`
`3, 4, 5, 6, 7`
`A uu B`
`1, 2, 3, 4, 5, 6`
`(A uu B) nn C`
`3, 4, 5, 6`
`A nn C`
`3, 5`
`B nn C`
`4, 6`
`(A nn C) uu (B nn C)`
`3, 4, 5, 6`
`:. (A uu B) nn C = (A nn C) uu (B nn C)` (সত্যতা যাচাই হলো)বিকল্প সমাধান:
দেওয়া আছে,
`A = {1, 3, 5} B = {2, 4, 6}`
এবং `C = {3, 4, 5, 6, 7}`
এখানে,
`A uu B = {1, 3, 5} uu {2, 4, 6}`
`= {1, 2, 3, 4, 5, 6}`
`:. (A uu B) nn C = {1, 2, 3, 4, 5, 6} nn {3, 4, 5, 6, 7}`
`= {3, 4, 5, 6}`
আবার,
`A nn C = {1, 3, 5 } nn {3, 4, 5, 6, 7}`
`=  {3, 5}`
`B nn C = {2, 4, 6} nn {3, 4, 5, 6, 7}`
`=  {4, 6}`
`:. (A nn C) uu (B nn C) = {3, 5} uu {4, 6}`
`= {3, 4, 5, 6}`
`:. (A uu B) nn C = (A nn C) uu (B nn C)`  (সত্যতা যাচাই হলো)
(iv).`(A nn B) uu C = (A uu C) nn (B uu C)`
সেট
উপাদান
`A`
`1, 3, 5`
`B`
`2, 4, 6`
`C`
`3, 4, 5, 6, 7`
`A nn B`
`O/`
`(A nn B) uu C`
`3, 4, 5, 6, 7`
`A uu C`
`1, 3, 4, 5, 6, 7`
`B uu C`
`2, 3, 4, 5, 6, 7`
`(A uu C) nn (B uu C)`
`3, 4, 5, 6, 7`
`:. (A nn B) uu C = (A uu C) nn (B uu C)`  (সত্যতা যাচাই হলো)বিকল্প সমাধান:
দেওয়া আছে,
`A = {1, 3, 5} B = {2, 4, 6}`
এবং `C = {3, 4, 5, 6, 7}`
এখানে,
`A nn B = {1, 3, 5} nn {2, 4, 6}`
`= {}`
`:. (A nn B) uu C = {} uu {3, 4, 5, 6, 7}`
`= {3, 4, 5, 6, 7}`
আবার,
`A uu C = {1, 3, 5} uu {3, 4, 5, 6, 7}`
`= {1, 3, 4, 5, 6, 7}`
`B uu C = {2, 4, 6} uu {3, 4, 5, 6, 7}`
`= {2, 3, 4, 5, 6, 7}`
`:. (A uu C) nn (B uu C) = {1, 3, 4, 5, 6, 7} nn {2, 3, 4, 5, 6, 7}`
`= {3, 4, 5, 6, 7}`
`:. (A nn B) uu C = (A uu C) nn (B uu C)`  (সত্যতা যাচাই হলো)

সমাধান:
দেওয়া আছে,
`Q = {x, y}` এবং `R = {m, n, l}`
এখানে,
`Q`এর উপসেটসমূহ `{x, y}, {x},{y}` এবং `O/`.
`:. P(Q) = {{x, y}, {x}, {y}, O/}`
আবার,
`R`এর উপসেটসমূহ `{m, n, l}, {m, n}, {m, l}, {n, l}, {m}, {n}, {l}` এবং `O/`.  
`:. P(R) = {{m, n, l}, {m, n}, {m, l}, {n, l}, {m}, {n}, {l}, O/}`    (Ans.)

সমাধান:
দেওয়া আছে,
`A = {a, b}, B = {a, b, c}`
`:. C = A uu B`
`= {a, b} uu {a, b, c}`
`= {a, b, c}`
এখানে,
`C` এর উপাদান সংখ্যা `3`.
এবং `C` এর উপসেটসমূহ `{a, b, c}, {a, b}, {b, c}, {c, a}, {a}, {b}, {c}, O/`.
`:. P(C) = {{a, b, c}, {a, b}, {b, c}, {c, a}, {a}, {b}, {c}, O/}`
`:. P(C)` = এর উপাদান সংখ্যা `8`
`= 2^3` 
`:. P(C)` এর উপাদান সংখ্যা `2^n`
যেখানে `n` হলো `C` এর উপাদান সংখ্যা।  (দেখানো হলো)