Binary Operations

Let $*$ be a binary operation on the set $Q$ of rational numbers. Find which of the operations has identity.

State whether the given statement is true or false and Justify.
If $*$ is a commutative binary operation on $N,$ then $a*\left(b*c\right)=(c*b)*a$.

State whether the given statement is true or false. Justify.

For an arbitrary binary operation $*$ on a set $N, a*a=a \forall a\in N.$

Number of binary operations on the set $\left\{a,b\right\}$ are

Consider a binary operation $*$ on $N$ defined as $a*b=a^3+b^3$. Choose the correct answer.

Define a binary operation $*$ on the set $\left\{0,1,2,3,4,5\right\}$ as
$a*b=\left\{\begin{array}{ll}a+b& \text{if} a+b<6\\ a+b-6& \text{if} a+b\ge 6\end{array}\right.$
Show that zero is the identity for this operation and each element $a\ne 0$ of the set is invertible with $6-a$ being the inverse of $a.$

Given a non‐empty set $X,$ let $*$ : $P\left(X\right)\times P\left(X\right)\to P\left(X\right)$ be defined as $A*B=(A-B)\cup (B-A),{\forall }^{}A,B\in P\left(X\right)$. Show that the empty set $\varphi$ is the identity for the operation $*$ and all the elements $A$ of $P\left(X\right)$ are invertible with $A^{-1}=A$.

Consider the binary operations $*$: $R\times R\to R$ and $o$ : $R\times R\to R$ defined as $a*b=|a-b|$ and $a o b=a,\forall a,b\in R.$ Show that $*$ is commutative but not associative, $o$ is associative but not commutative. Further, show that $\forall a,b,c\in R,a*(b o c)=\left(a*b\right) o \left(a*c\right)$. [If it is so, we say that the operation $*$ distributes over the operation $o$]. Does $o$ distribute over $*$? Justify your answer.

Given a non‐empty set $X,$ consider the binary operation $*$ : $P\left(X\right)\times P\left(X\right)\to P\left(X\right)$ given by $A*B=A\cap B \forall A,B$ in $P\left(X\right)$, where $P\left(X\right)$ is the power set of $X$. Show that $X$ is the identity element for this operation and $X$ is the only invertible element in $P\left(X\right)$ with respect to the operation $*.$

Let $A=N\times N$ and $*$ be the binary operation on $A$ defined by $\left(a,b\right)*\left(c,d\right)=\left(a+c,b+d\right).$ Show that $*$ is commutative and associative. Find the identity element for $*$ on $A$, if any.

 Let $*$ be a binary operation on the set $Q$ of rational numbers as $a*b=a{b}^2.$ Find whether the binary operation is commutative or associative or both.

Let $*$ be a binary operation on the set $Q$ of rational numbers as $a*b=\frac{ab}{4}.$ Find whether the binary operation is commutative or associative or both.

Let $*$ be a binary operation on the set $Q$ of rational numbers as $a*b=(a-b{)}^2.$ Find whether the binary operation is commutative or associative or both.

Let $*$ be a binary operation on the set $Q$ of rational numbers as $a*b=a+ab.$ Find whether the binary operation is commutative or associative or both.

Let $*$ be a binary operation on the set $Q$ of rational numbers as $a*b=a^2+b^2.$ Find whether the binary operation is commutative or associative or both.

Let $*$ be a binary operation on the set $Q$ of rational numbers be defined as $a*b=a-b.$ Find whether the binary operation is commutative or associative or both.

Let $*$ be the binary operation on $N$ defined by $a*b=H.C.F.$of $a$ and $b.$ Is $*$ commutative? Is $*$ associative? Does there exist identity for this binary operation on $N$?

Is $*$ defined on the set $\left\{1,2,3,4,5\right\}$ by $a*b=L.C.M$. of $a$ and $b$ a binary operation? Justify your answer.