Sets and Relations

Let $A$ and $B$ be two sets such that $A\cap\,X=B\cap\,X=\varphi\,$ and $A\cup\,X=B\cup\,X$ for same set $X$. Then,

If$R=\left{\left(x,y\right):x,y\in\,Z,x^{2}+3y^{2}\leq\,8\right}$is a relation on the set of integers$Z$, then the domain of$R^{-1}$is

Let $P\left(S\right)$ denote the power set of $S=\left{1,2,3,\ldots\,,10\right}$. Define the relations $R_{1}$and $R_{2}$on $P\left(S\right)$ as $AR_{1}B$ if…

Let $Z$ be the set of all integers, $A=\left{(x,y)\in\,Z\times\,Z:(x-2)^{2}+y^{2}\leq\,4\right}$ $B=\left{(x,y)\in\,Z\times\,Z:x^{2}+y^{2}\leq\,4\right}\text{and}$…

In a group of$100$persons$75$speak English and$40$speak Hindi. Each person speaks at least one of the two languages. If the number of persons who speak only English is$\alpha\,$and the number of persons who speaks only…

Let $A,B$ and $C$ be sets such that $\varphi\,\neq\,A\cap\,B\subseteq\,C.$ Then which of the following statements is not true?

Let $A=\left{0,3,4,6,7,8,9,10\right}$ and $R$ be the relation defined on $A$ such that $R\left{\left(x,y\right)\in\,A\times\,A:x-y\text{is odd positive integer}\text{or}x-y=2\right}$. The minimum number of elements that…

Let $A=\{1,2,3\}$. The number of relations on $A$, containing $(1,2)$ and $(2,3)$, which are reflexive and transitive but not symmetric, is ______ -

The number of symmetric relations defined on the set ${1,2,3,4}$ which are not reflexive is _______?

Let $A=\left{1,2,3,4\right}$ and $R=\left{(1,2),(2,3),(1,4)\right}$ be a relation on $A$. Let $S$be the equivalence relation on $A$such that $R\subset\,S$and the number of elements in $S$is $n$. Then, the minimum value…

Let$A=\left{1,2,3,.\operatorname{..20}\,\right}$. Let$R_{1}$and$R_{2}$two relation on $A$ such that $R_{1}={\left(a,b\right):b$is divisible by $a$} $R_{2}={\left(a,b\right):a$is an integral multiple of $b$} Then, number…

Let $R = \{(x, y) \in \mathbb{N} \times \mathbb{N} : \log_e(x + y) \leq 2\}$. Then the minimum number of elements, required to be added in $R$ to make it a transitive relation, is __________.