Properties of a Binary Operation

Let $A=\left\{1,2,4\right\}$. Define as operation $*$ by $a*b=\mathrm{HCF}\left(a,b\right)$. Prepare its composition table. Is $*$ a binary operation? Find the identity element in $A$.

Let $A=\left\{1,2,3\right\}$. Define as operation $*$ by $a*b=\mathrm{LCM}\left(a,b\right)$. Prepare its composition table. Is $*$ a binary operation?

If $*$ is a binary operation on $Z$ defined by $a*b=a+5b$, find the value of $3*2 \mathrm{and} 2*3$. Are they equal?

Examine the following is a binary operation $a*b=\frac{a+b}{2}, a,b\in Q$. For binary operation check the commutative and associative property.

Let $*$ be a binary operation in a set $Q$ of rational numbers given as $a*b={\left(2a-b\right)}^2, a,b\in Q$. Find $3*5 \mathrm{and} 5*3$. Is $3*5=5*3$?

Show that the binary operation $*:N\times N\to N$ defined as $a*b\to \mathrm{lcm}\left(a,b\right)$ is associative.

Consider the set $A=\left\{1, \omega , {\omega }^2\right\}$ of all cube tools of unity. Construct the composition table for multiplication on $A$. Find the identity element.

Let $A=\left\{1,-1,i,-i\right\}$ where $i=\sqrt{-1}$. Prepare a composition table for multiplication on $A$. Find the identity element.

Let $A=\left\{1,-1,i,-i\right\}$ where $i=\sqrt{-1}$. Prepare a composition table  for multiplication on $A$. Is it a binary operation?

Let $A=\left\{1,-1,i,-i\right\}$ where $i=\sqrt{-1}$. Prepare a composition table  for multiplication on $A$ and show that $A$ is closed for multiplication.

Consider the binary operation $*$ defined on the set of rational numbers $Q$ by $a*b=a-b+ab$. Show that $*$ is neither commutative nor associative.

Let $*$ be a binary operation on $R-\left\{-1\right\}$, defined by $a*b=\frac{a}{b+1}$. Show that $*$ is neither commutative nor associative.

Is binary operation $*$ defined on the set of all natural numbers $N$ by $a*b=\frac{a+b}{2}$, for all $a,b\in N$ is commutative. It it associative?

Let a binary operation $*$ on a set $A$ having more than one element, be defined by $a*b=a$ for $a,b\in A$. Show that $(A, *)$ is associative, but it has no identity element. Is $(A, *)$ commutative ?

Let a binary operation $*$ on $N$, the set of all natural numbers, be defined by $a*b=a^b$ for $a,b\in N$. Show that $(N, *)$ is neither commutative nor associative.

Let $Q$ be the set of non-zero rational numbers and $*$ be a Binary Operation on $Q$ defined by$a*b=\frac{ab}{2}$. Show that all the first five properties hold in $Q$ w.r.t. $*$.

Prove that in the set $\left\{1,-1\right\}$ with the multiplication as binary operation all the first five properties are satisfied.

Show that all the properties of Binary Operations hold in the set $I$ of all integers with respect to the operation of addition.

Let $A$ be set of all real numbers except $-1$ and operation $*$ be defined on $A$ as $a*b=a+b+ab$ for all $a,b\in A.$ 

The value of $x$ in the equation $1*x*3=5$ is

Consider a binary operation $*$ on $N$ defined as $a*b=1,$ for all $a,b\in N.$ Choose the correct answer.