A relation R is defined on the set N of natural numbers as xRy $$\Rightarrow$$ x² $$-$$ 4xy + 3y² = 0. Then, which one of the following is correct?

If $$A = \{ x \in Z:{x^3} - 1 = 0\} $$ and $$B = \{ x \in Z:{x^2} + x + 1 = 0\} $$, where, Z is set of complex numbers, then what is A $$\cap$$ B equal to?

Consider the following statements for the two non-empty sets A and B.

1. $$(A \cap B) \cup (A \cap \overline B ) \cup (\overline A \cap B) = A \cup B$$

2. $$(A \cup (\overline A \cap \overline B )) = A \cup B$$

Which of the above statements is/are correct?

Let X be a non-empty set and let A, B, C be subsets of X. Consider the following statements.

1. $$A \subset C \Rightarrow (A \cap B) \subset (C \cap B),\,(A \cup B) \subset (C \cup B)$$

2. $$(A \cap B) \subset (C \cap B)$$ for all sets $$B \Rightarrow A \subset C$$

3. $$(A \cup B) \subset (C \cup B)$$ for all sets $$B \Rightarrow A \subset C$$

Which of the above statements are correct?