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Let $\mathrm{X}$ be a non-empty set and let $\mathrm{A}, \mathrm{B}, \mathrm{C}$ be subsets of $\mathrm{X}$. Consider the following statments:
(1) $\mathrm{A} \subset \mathrm{C} \Rightarrow(\mathrm{A} \cap \mathrm{B}) \subset(\mathrm{C} \cap \mathrm{B})(\mathrm{A} \cup \mathrm{B}) \subset(\mathrm{C} \cap \mathrm{B})$
(2) $(\mathrm{A} \cup \mathrm{B}) \subset(\mathrm{C} \cap \mathrm{B})$ for all sets $\mathrm{B} \Rightarrow \mathrm{A} \subset \mathrm{C}$
(3) $(\mathrm{A} \cup \mathrm{B}) \subset(\mathrm{C} \cup \mathrm{B})$ for all sets $\mathrm{B} \Rightarrow \mathrm{A} \subset \mathrm{C}$
Which of the above statements are correct?
Options:
(1) $\mathrm{A} \subset \mathrm{C} \Rightarrow(\mathrm{A} \cap \mathrm{B}) \subset(\mathrm{C} \cap \mathrm{B})(\mathrm{A} \cup \mathrm{B}) \subset(\mathrm{C} \cap \mathrm{B})$
(2) $(\mathrm{A} \cup \mathrm{B}) \subset(\mathrm{C} \cap \mathrm{B})$ for all sets $\mathrm{B} \Rightarrow \mathrm{A} \subset \mathrm{C}$
(3) $(\mathrm{A} \cup \mathrm{B}) \subset(\mathrm{C} \cup \mathrm{B})$ for all sets $\mathrm{B} \Rightarrow \mathrm{A} \subset \mathrm{C}$
Which of the above statements are correct?
Solution:
1627 Upvotes
Verified Answer
The correct answer is:
2 and 3 only
Statements $(2)$ and $(3)$ are correct.
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