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Let `A`, `B` and `C` be the sets such that `A uu B=A uu C` and `A nn B = A nn C`. Show that `B=C` |
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Answer» (i) `because` Sets A, B and C are such that `A cup B = A cup C` or `(A cup B) cap B = (A cupC) cap B` or `(A cap B) cup (B cap B) = (A cap B) cup (C cap B)` (From distributive law) or `B = (A cap B) cup (C cap B)` ….(1) [Since `(A cap B) cup (B cap B)=B`] Again `(A cup B) = (A cup C)` or `(A cup B) cap C = (A cup C) cap C` or `(A cap C) cup (B cap C)= (A cap C) cup (C cap C)` [From distributive law) or `(A cap C) cup (B cap C) = C` [Since `(A cap B) cup (C cap C)=C`] then `C = (A cap C) cup (B cup C)`...(2) `because` Given `A cap B = A capC` then, replace `(A cap C)` by `(A cap B)` in equation (2), `C = (A cap B) cup (B cup C)`...(3) Now comparing equation (1) and equation (3), `B=C`. |
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