Basics of Relations

In a set of $5$ numbers the sum of two of these numbers is $6$ more than the sum of the remaining three numbers, whereas the sum of these three numbers is two times one of those two numbers. What definitely is one of those two numbers?

The binary operation $*$ on the set $\mathrm{\mathbb{Z}}$ of integers defined by $a*b=a^2+b^2$ is

The binary operation $*:R\times R\to R$ defined by $*(x, y)=\max \{x, y\}$ is

A relation $R$ is defined in the set $Z$ of integers as follows $\left(x, y\right)\in R$ if and only if $x^2+y^2=9$. Which of the following is false?

$A=\left\{3,9\right\}$, $B=\left\{9,6\right\}$ and $C=\left\{3,6\right\}$. Verify associative property of Cartesian product of sets.

$A=\left\{1,4\right\}$, $B=\left\{4,5\right\}$ and $C=\left\{1,5\right\}$. Verify associative property of Cartesian product of sets.

$A=\left\{1,2\right\}$, $B=\left\{2,3\right\}$ and $C=\left\{1,3\right\}$. Verify associative property of Cartesian product of sets.

If $A=\left\{5,9\right\}$, $B=\left\{6,8\right\}$ and $C=\left\{9,7\right\}$, then prove that, $A\times (B\cup C)=(A\times B)\cup (A\times C)$.

If $A=\left\{b,a\right\}$, $B=\left\{c,e\right\}$ and $C=\left\{a,d\right\}$, then prove that, $A\times (B\cup C)=(A\times B)\cup (A\times C)$.

If $A=\left\{0,1\right\}$, $B=\left\{2,4\right\}$ and $C=\left\{1,3\right\}$, then prove that, $A\times (B\cup C)=(A\times B)\cup (A\times C)$.

What can be the domain as range from the following diagram of relation?

There are two sets a and b and $A\times B$ contains $6$ Elements of three elements of $A\times B$ as.

$\left(5,6\right), \left(2,9\right) \& \left(6,8\right)$ Then which one is not the elements of $\left(A\times B\right).$

Let number of elements in set $P$ is $3$. If $a,b,c$ are the number of identity, reflexive and symmetric relations on set $P$, then the value of $a+b+c$ is equal to

If a relation $R$ defined on the set of integers $I$ such that $R=\left\{\left(x,y\right):2x^2+3y^2\le 15, x,y\in I\right\}$, then the range of $R^{-1}$ is

For two given non-empty sets $A$ and $B$, $A\cap (A\cap B{)}^{\text{'}}$ is equal to

Let $A$ be a set consisting of $10$ elements. The number of non-empty relations from $A$ to $A$ that are reflexive but not symmetric is

The number of ordered pairs $\left(a,b\right)$ of positive integers such that $\frac{2a-1}{b}$ and $\frac{2b-1}{a}$ are both integers is 

Which of the following is a ordered $3-$ tuple?