Suppose \( X = \{1, 2, 3, 4\} \) and R is a relation on X. If \( R = \{(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (2, 3), (3, 2)\} \), then which one of the following is correct?
A.
R is reflexive and symmetric, but not transitive
B.
R is symmetric and transitive, but not reflexive
C.
R is reflexive and transitive, but not symmetric
D.
R is neither reflexive nor transitive, but symmetric
D. R is reflexive, but neither symmetric nor transitive
4.
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. \( (A \cap B) \cup (A \cap \bar{B}) \cup (\bar{A} \cap B) = A \cup B \)
2. \( A \cup (\bar{A} \cap B) = A \cup B \)
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. \( A \subseteq C \Rightarrow (A \cap B) \subseteq (C \cap B), (A \cup B) \subseteq (C \cup B) \)
2. \( (A \cap B) \subseteq (C \cap B) \) for all sets \( B \Rightarrow A \subseteq C \)
3. \( (A \cup B) \subseteq (C \cup B) \) for all sets \( B \Rightarrow A \subseteq C \)
Which of the above statements are correct?
