MCQ_SINGLE

If R is the smallest equivalence relation on the set ${1, 2, 3, 4}$ such that ${((1, 2), (1, 3))} \subset R$, then the number of elements in $R$ is __________.

NUMERICAL

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 __________.

MCQ_SINGLE

Let a relation $R$ on $N \times N$ be defined as: $(x_1, y_1) R (x_2, y_2)$ if and only if $x_1 \le x_2$ or $y_1 \le y_2$. Consider the two statements: (I) $R$ is reflexive but not symmetric. (II) $R$ is transitive Then which one of the following is true?

NUMERICAL

For $n \geq 2$, let $S_n$ denote the set of all subsets of $\{1,2, \ldots, n\}$ with no two consecutive numbers. For example $\{1,3,5\} \in S_6$, but $\{1,2,4\} \notin S_6$. Then $n\left(S_5\right)$ is equal to ________

NUMERICAL

Let $A=\{0,3,4,6,7,8,9,10\}$ and $R$ be the relation defined on $A$ such that $R=\{(x, y) \in A \times A: x-y$ 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 ____________.

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