Embibe Experts Solutions for Chapter: Set Theory and Relations, Exercise 1: Exercise 1

What is the number of natural numbers less than or equal to $1000$ which are neither divisible by $10$ nor $15$ nor $25$?

A school awarded $38$ medals in football, $15$ in basketball and $20$ in cricket. Suppose these medals went to a total of $58$ students and only three students got medals in all three sports. If only $5$ students got medals in football and basketball, then the number of medals received in exactly two of the three sports is

If we define a relation $R$ on the set $N\times N$ as $(a,b)R(c,d)\iff a+d=b+c$ for all $\left(a,b\right),\left(c,d\right)\in N\times N$, then the relation is

Let $Q$ be the set of all rational numbers and $R$ be the relation defined as $R=\left\{\left(x,y\right):1+xy>0;x,y\in Q\right\}$ The relation $R$ is

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

If $A=\{x: x$ is a multiple of $3\}$ and $B=\{x: x$ is a multiple of $4\}$ and $C=\{x: x$ is a multiple of $12\}$, then which one of the following is a null set?

If $A=\left\{x\in R: x^2+6x-7<0\right\}$ and $B=\left\{x\in R: x^2+9x+14>0\right\}$, then which of the following is/are correct?

$\mathrm{I}.$$\left(A\cap B\right)=\left(-2,1\right)$                $\mathrm{II}.$ $\left(A\backslash B\right)=\left(-7,-2\right)$

Select the correct answer using the code given below 

$A, B, C$ and $D$ are four sets such that $A\cap B=C\cap D=\mathrm{\varphi }$. Consider the following

$\mathrm{I}.$ $A\cup C$ and $B\cup D$ are always disjoint

$\mathrm{II}.$ $A\cap C$ and $B\cap D$ are always disjoint

Which of the above statement(s) is / are correct?