6 questions from 2022
For \( \alpha \in \mathbb{N} \), consider a relation \( R \) on \( \mathbb{N} \) given by \( R=\{(x,y)\}: 3x+\alpha y \text{ is a multiple of } 7\} \). The relation \( R \) is an equivalence relation if and only if:
Let \( A = \{1,2,3,4,5,6,7\} \) and \( B = \{3,6,7,9\} \). Then the number of elements in the set \( \{ C \subseteq A : C \cap B \neq \phi \} \) is ______.
Let \( A = \{1, 2, 3, 4, 5, 6, 7\} \). Define \( B = \{T \subseteq A : \) either \( 1 \notin T \) or \( 2 \in T\} \) and \( C = \{T \subseteq A : T \) the sum of all the element of \( T \) is a prime number\( \} \). Then the number of element in the set \( B \cup C \) is \( \_ \_ \_ \_ \_ \_ \)
The probability that a relation \( R \) from \( \{x,y\} \) to \( \{x,y\} \) is both symmetric and transitive, is equal to:
Let a set \( A = A_1 \cup A_2 \cup \ldots \cup A_k \), where \( A_i \cap A_j = \phi \) for \( i \neq j \), \( 1 \le i, j \le k \). Define the relation \( R \) from \( A \) to \( A \) by \( R = \{(x,y): y \in A_i \text{, if and only if } x \in A_i, 1 \le i \le k \} \). Then, \( R \) is:
Let \( A = \{ x \in R : |x + 1| < 2 \} \) and \( B = \{ x \in R : |x - 1| \ge 2 \} \). Then which one of the following statements is NOT true?
