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(i) If x in A and A in B , "then "x inB....

(i) If `x in A and A in B , "then "x inB`.
(ii) If `A subBandB inC, "then" A in C`.
(iii) If A `subB andBsubC,"then" AsubC`.
(iv) If A `notinBandBnotinC,"then"AnotinC`.
(v)If `x notinAandAnotinB, "then"x inB`.
(vi) If A `subBandx notinB, "then"x notinA`. Find true and false.

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The correct Answer is:
To determine the truth values of the given statements, we will analyze each statement step by step using set theory concepts. ### Step-by-Step Solution: 1. **Statement (i): If `x in A and A in B`, then `x in B`.** - **Analysis**: - If `x` is an element of set `A`, and set `A` is a subset of set `B`, it does not guarantee that `x` is in `B`. - For example, if `A` is a subset of `B` but `B` has additional elements not in `A`, `x` could be an element of `A` but not in `B`. - **Conclusion**: **False** 2. **Statement (ii): If `A ⊆ B and B in C`, then `A in C`.** - **Analysis**: - If `A` is a subset of `B`, and `B` is a subset of `C`, then all elements of `A` are also in `C`. - **Conclusion**: **True** 3. **Statement (iii): If `A ⊆ B and B ⊆ C`, then `A ⊆ C`.** - **Analysis**: - This follows the transitive property of subsets. If `A` is a subset of `B`, and `B` is a subset of `C`, then `A` must also be a subset of `C`. - **Conclusion**: **True** 4. **Statement (iv): If `A notin B and B notin C`, then `A notin C`.** - **Analysis**: - Just because `A` is not an element of `B` and `B` is not an element of `C`, it does not imply that `A` cannot be an element of `C`. - For example, `A` could be a separate set that is not related to `B` or `C`. - **Conclusion**: **False** 5. **Statement (v): If `x notin A and A notin B`, then `x in B`.** - **Analysis**: - If `x` is not in `A` and `A` is not in `B`, it does not imply that `x` must be in `B`. `x` could be outside both sets. - **Conclusion**: **False** 6. **Statement (vi): If `A ⊆ B and x notin B`, then `x notin A`.** - **Analysis**: - If `A` is a subset of `B`, and `x` is not in `B`, then `x` cannot be in `A` either, since all elements of `A` must be in `B`. - **Conclusion**: **True** ### Summary of Truth Values: 1. (i) False 2. (ii) True 3. (iii) True 4. (iv) False 5. (v) False 6. (vi) True
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