Recent questions and answers in Set Theory & Algebra

$A=\{0,1,2,3, \ldots\}$ is the set of non-negative integers. Let $F$ be the set of functions from $A$ to itself. For any two functions, $f_{1}, f_{2} \in \mathrm{~F}$, we...

Let $R$ be a binary relation on the set $\{1,2, \ldots, 10\}$, where $(x, y) \in R$ if the product of $x$ and $y$ is square of an integer. Which of the following properti...

The number of bijections $f(\cdot)$ from the set $S=\{1,2,3,4\}$ to itself such that $f(f(n))=n$, for all $n \in S$, is $\_\_\_\_\_\_\_$. (Answer in integer)

Consider the following relations on $\{1,2,3,4\}$. Which of the following relations are reflexive?$\mathrm{R}_{1}=\{(1,1),(1,2),(2,1),(2,2),(3,4),(4,1),(4,4)\}$$\mathrm{R...

Let $Z_{n}$ be the group of integers $\{0,1,2, \ldots, n-1\}$ with addition modulo $n$ as the group operation. The number of elements in the group $Z_{2} \times Z_{3} \ti...

​​Let $\mathcal{F}$ be the set of all functions from $\{1, \ldots, n\}$ to $\{0,1\}$. Define the binary relation $\leqslant$ on $\mathcal{F}$ as follows:$\forall f, g \in...

Which of the following is a cyclic group?The set of all integers under addition.The set of all real numbers under addition.The set of all integers under multiplication.Th...

Let $A$ be the set of all finite binary strings, including the empty string $\epsilon$. Define $F:A\to A$ as follows: $F(w)$ is obtained by writing $0$ before the string ...

Let $A$ and $B$ be non-empty sets and let $f:A\to B$ be a function. For $S\subseteq A$, define $S^c=A-S$. For $Y\subseteq B$, define $Y^c=B-Y$. Which of the following sta...

Find the domain of the function $F(x)=\sqrt{\ln(x-1)}+\ln\left(\dfrac{\sqrt{x+1}-2}{x^2-5x+6}\right)$.$[2,\infty)$ $(2,\infty)$ $(2,3)\cup(3,\infty)$ $[2,3)\cup(3,\infty)...

Let $A=\{1,2,3,4,5,6\}$. Find the number of bijections $f:A\to A$ such that $f(1)\neq1$, $f(2)\neq2$, $f(3)\neq3$, and $f(4)=5$ if and only if $f(5)=4$.

Let $A$ be the set of all non-constant linear functions from $\mathbb R$ to $\mathbb R$. Define $T:A\to\mathbb R$ by $T(\phi)=$ the unique real number $x$ for which $\phi...

Let $F$ and $G$ be Boolean functions of degree $n$, where the order relation is defined by $F\leq G$ if and only if $F(x)\leq G(x)$ for every $x\in{0,1}^n$. Also define $...

A $4$-ary boolean function is a function $f:\{0,1\}^4\to\{0,1\}$. Let $h$ be a fixed $4$-ary boolean function. We say that another $4$-ary boolean function $g$ is strongl...