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The number of elements in the set \( \{\, n \in \mathbb{Z} : |n^2 - 10n + 19| < 6 \,\} \) is __________.

Let \( A=\{0,3,4,6,7,8,9,10\} \) and \( R \) be the relation defined on \( A \) such that \( R=\{(x,y)\in A\times A:x-y \text{ is odd positive integer or } x-y=2\} \). The minimum number of elements that must be added to the relation \( R \), so that it is a symmetric relation, is equal to ________.

The minimum number of elements that must be added to the relation \( R=\{(a,b),(b,c),(b,d)\} \) on the set \( \{a,b,c,d\} \) so that it is an equivalence relation, is ________ .

Let \( A=\{1,2,3,4,5,6,7\} \). Then the relation \( R=\{(x,y)\in A\times A: x+y=7\} \) is:

Among the relations:

Let \( A = \{1, 2, 3, 4, \ldots, 10\} \) and \( B = \{0, 1, 2, 3, 4\} \). The number of elements in the relation \( R = \{(a, b) \in A \times A : 2(a-b)^2 + 3(a-b) \in B\} \) is ______________.

Let \( R \) be a relation defined on \(\mathbb{N}\) as \( a \, R \, b \) if \( 2a + 3b \) is multiple of \( 5 \), \( a, b \in \)\(\mathbb{N}\) . Then \( R \) is:

Let the number of elements in sets \( A \) and \( B \) be five and two respectively. Then the number of subsets of \( A \times B \) each having at least \( 3 \) and at most \( 6 \) elements is:

Let \( R \) be a relation on \( \mathbb{R} \), given by \( R=\left\{(a,b):3a-3b+\sqrt{7}\ \text{ is an irrational number}\right\} \).
Then \( R \) is :

An organization awarded \( 48 \) medals in event ‘\( A \)’, \( 25 \) in event ‘\( B \)’ and \( 18 \) in event ‘\( C \)’. If these medals went to total \( 60 \) men and only \( 5 \) men got medals in all the three events, then, how many received medals in exactly two of three events?

Let \( P(S) \) denote the power set of \( S=\{1,2,3,\ldots,10\} \). Define the relations \( R_{1} \) and \( R_{2} \) on \( P(S) \) as \( A R_{1} B \) if \( (A\cap B^{c})\cup(B\cap A^{c})=\varnothing \) and \( A R_{2} B \) if \( A\cup B^{c}=B\cup A^{c} \), \( \forall A,B\in P(S) \). Then:

Let \( A = \{-4, -3, -2, 0, 1, 3, 4\} \) and \( R = \{(a, b) \in A \times A : b = |a| \text{ or } b^2 = a + 1\} \) be a relation on \( A \). Then the minimum number of elements, that must be added to the relation \( R \) so that it becomes reflexive and symmetric, is _________.

The minimum number of elements that must be added to the relation \( R=\{(a,b),(b,c)\} \) on the set \( \{a,b,c\} \) so that it becomes symmetric and transitive is:

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JEE Main 2023 10th April Morning Shift

The number of elements in the set \( \{\, n \in \mathbb{Z} : |n^2 - 10n + 19| < 6 \,\} \) is __________.