৪ u 1 2 3 4 5 6 7 a b, সেট ও ফাংশন, গণিত

Question:৪. `u = {1, 2, 3, 4, 5, 6, 7}, A = {1, 3, 5},  B = {2, 4, 6}`এবং`C = {3, 4, 5, 6, 7}` হলে, নিম্নলিখিত ক্ষেত্রে সত্যতা যাচাই কর:

(i). `(A uu B)' = A' nn B'`

(ii). `(B nn C)' = B' uu C'`

(iii). `(A uu B) nn C = (A nn C) uu (B nn C)`

(iv). `(A nn B) uu C = (A uu C) nn (B uu C)`

Answer 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)

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৪. `u = {1, 2, 3, 4, 5, 6, 7}, A = {1, 3, 5}, B = {2, 4, 6}`abong`C = {3, 4, 5, 6, 7}` hole, nimonlikhit kkhetre shottata jachai karo: (i). `(A uu B)' = A' nn B'` (ii). `(B nn C)' = B' uu C'` (iii). `(A uu B) nn C = (A nn C) uu (B nn C)` (iv). `(A nn B) uu C = (A uu C) nn (B uu C)`

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`