Sets and Relations

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 survey of 220 students of a higher secondary school, it was found that at least 125 and at most 130 students studied Mathematics; at least 85 and at most 95 studied Physics; at least 75 and at most 90 studied…

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

A group of $40$ students appeared in an examination of $3$ subjects - Mathematics, Physics & Chemistry. It was found that all students passed in at least one of the subjects, $20$ students passed in Mathematics, $25$…

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?

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 $P\left(S\right)$ 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…

The number of elements in the set$S=\left{\left(x,y,z\right):x,y,z\in\,Z,x+2y+3z=42,x,y,z\geq\,0\right}$equals ________

Let $A=\left{1,2,3,.\operatorname{...100}\,\right}$. Let$R$be a relation on $A$ defined by$\left(x,y\right)\in\,R$if and only if$2x=3y$. Let $R_{1}$ be a symmetric relation on $A$ such that$R\subset\,R_{1}$and the…

Let $A = \{2, 3, 4, 5, 6\}$. Let R be a relation on the set $A \times A$ given by $(x, y) R (z, w)$ if and only if $x$ divides $z$ and $y \leq w$. Then the number of elements in R is _______.

Let $X=\left\{n\in\,N:1\leq\,n\leq\,50\right\}$. If $A=\left\{n\in\,X:nisamultipleof2\right\}$ and $B=\left\{n\in\,X:nisamultipleof7\right\}$, then the number of elements in the smallest subset of $X$, containing both…

Let $A=\left{n\in\,N:H.C.F.\left(n,45\right)=1\right}$and let $B=\left{2k:k\in\,\left{1,2,\ldots\,,100\right}\right}$. Then the sum of all the elements of $A\cap\,B$is _____.