Sets and Relations

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

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

Let $A=\left{1,2,3,4,5,6,7\right}$. Define $B=${$T\subseteq\,A$: either $1\notin\,T$or $2\in\,T$}and $C=${$T\subseteq\,A:T$the sum of all the elements of$T$ is a prime number.} Then the number of elements in the set…

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 _____.

Let $A=\left{1,a_{1},a_{2}\ldots\,\ldots\,a_{18},77\right}$be a set of integers with $1<a_{1}<a_{2}<\ldots\,..<a_{18}<77$. Let the set $A+A=\left{x+y:x,y\in\,A\right}$contain exactly $39$elements. Then, the value of…

Paragraph: Read the following information and answer: Consider the following Venn diagram, where $X, Y$ and $Z$ are three sets. Let the number of elements in $Z$ be denoted by $n(Z)$ which is equal to 90 . Question:…

For any three non-empty sets $\mathrm{A}, \mathrm{B}, \mathrm{C}$ what is $(\mathrm{A} \cup \mathrm{B})-\{(\mathrm{A}-\mathrm{B}) \cup(\mathrm{B}-\mathrm{A}) \cup(\mathrm{A} \cap \mathrm{B})\}$ equal to?

Consider the following in respect of sets A and B : $(\mathrm{A}-\mathrm{B}) \cup \mathrm{B}=\mathrm{A}$ $(\mathrm{A}-\mathrm{B}) \cup \mathrm{A}=\mathrm{A}$ $(A-B) \cap B=\phi$ $A \subseteq B \Rightarrow A \cup B=B$…

In a class of 60 students, 45 students like music, 50 students like dancing, 5 students like neither. Then the number of students in the class who like both music and dancing is

Let $A=(x \in R:-1 \leq x \leq 1)$ and $S$ be the subset of $A \times B$, defined by $S=\left[(x, y) \in A \times B: x^2+y^2=1\right]$ Which one of the following is correct?

In a state with a population of $75 \times 10^6, 45 \%$ of them know Hindi, 22% know English, 18% know Sanskrit, 12% know Hindi and English, 8% know English and Sanskrit, 10% know Hindi and Sanskrit and 5% know all the…