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

Which one of the following is a null set?

In an examination out of $100$ students, $75$ passed in English, $60$ passed in Mathematics and $45$ passed in both English and Mathematics. What is the number of students passed in exactly one of the two subjects?

If $A$ and $B$ are subsets of a set $X$, then what is $\left\{A\cap \left(X-B\right)\right\}\cup B$ equal to?

If $A=\left\{1, 2, 5, 6\right\}$ and $B=\left\{1, 2, 3\right\}$, then what is $\left(A\times B\right)\cap \left(B\times A\right)$ equal to?

Let $R=\{x\mid x\in N, x$ is a multiple of $3$ and $x\le 100\}$, $S=\{x\mid x\in N, x$ is a multiple of $5$ and $x\le 100\}$. What is the number of elements in $\left(R\times S\right)\cap \left(S\times R\right)?$

If $A=\left\{a, b, c\right\}$ and $R=\left\{\left(a, a\right), \left(a, b\right), \left(b, c\right), \left(b, b\right), \left(c, c\right), \left(c, a\right)\right\}$ is a binary relation on $A$, then which one of the following is correct?

For real numbers $x$ and $y$, define a relation $R, xRy$ if and only if $x-y+\sqrt{2}$ is an irrational number. Then, the relation $R$ is

Let $S$ denotes a set of all integers. Define a relation $R$ on $S$ as $\text{'}aRb\text{'}$ if $ab\ge 0$, where $a, b\in S$. Then, $R$ is