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If A , B, C are any three sets , then th...

If A , B, C are any three sets , then the correct match of the following is :
`{:("Column -I",,"Column -II"),((i) Auu(BcapC),,(a)(AcapB)uu(AcapC)),((ii)Acap(BuuC),,(b)(AuuB)cap(AuuC)),((iii)Auu(BuuC),,(c)(AcapB)capC.),((iv)Acap(BcapC),,(d)(AuuB)uuC.):}`

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem of matching the expressions from Column I with those from Column II, we will analyze each expression step by step. ### Step-by-Step Solution: 1. **Expression (i): A ∪ (B ∩ C)** - **Find B ∩ C**: This represents the elements common to both sets B and C. - **Assuming elements**: Let's say B = {D, B, E, G} and C = {C, E, G, F}. - **Common elements**: The intersection B ∩ C = {E, G}. - **Union with A**: Now, if A = {A, D, C, E, G, F}, then A ∪ (B ∩ C) = {A, D, C, E, G, F}. - **Result**: This matches with (d) in Column II. 2. **Expression (ii): A ∩ (B ∪ C)** - **Find B ∪ C**: This represents all elements in either B or C. - **Union of B and C**: B ∪ C = {D, B, E, G, C, E, G, F} = {B, C, D, E, F, G}. - **Intersection with A**: A ∩ (B ∪ C) = {A, D, C, E, G, F} ∩ {B, C, D, E, F, G} = {D, G, F}. - **Result**: This matches with (a) in Column II. 3. **Expression (iii): A ∪ (B ∪ C)** - **Find B ∪ C**: As calculated above, B ∪ C = {B, C, D, E, F, G}. - **Union with A**: A ∪ (B ∪ C) = {A, D, C, E, G, F} ∪ {B, C, D, E, F, G} = {A, B, C, D, E, F, G}. - **Result**: This matches with (c) in Column II. 4. **Expression (iv): A ∩ (B ∩ C)** - **Find B ∩ C**: From earlier, B ∩ C = {E, G}. - **Intersection with A**: A ∩ (B ∩ C) = {A, D, C, E, G, F} ∩ {E, G} = {G}. - **Result**: This matches with (b) in Column II. ### Final Matching: - (i) A ∪ (B ∩ C) → (d) - (ii) A ∩ (B ∪ C) → (a) - (iii) A ∪ (B ∪ C) → (c) - (iv) A ∩ (B ∩ C) → (b) ### Summary of Matches: - (i) → (d) - (ii) → (a) - (iii) → (c) - (iv) → (b)
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