Let * be the binary operation on N defined by, a * b = H.C.F. of … - Vidyadip

Question

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

✓

Answer

The binary operation * on N is defined as:
a * b = H.C.F. of a and b.
it is known that:
H.C.F. of a and b = H.C.F. of b and a. $\text{a, b}\in\text{N}$.
Therefore, a * b = b * a
Thus, the operation * is commutative.
For $\text{a, b, c}\in\text{N}$, we have:
(a * b) * c = (H.C.F of a and b) * c = H.C.F. of a, b and c
a * (b * c) = a * (H.C.F. of b and c) = H.C.F. of a, b and c
Therefore, (a * b) * c = a * (b * c)
Thus, the operation * is associative.
Now, an element $\text{e}\in\text{N}$ will be the identity for the operation
if a * e = a = e * a, $\forall\text{ a}\in\text{N}$.
But this relation is not true for any $\text{a}\in\text{N}$.
Thus, the operation * does not have any identity in N.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "5 Marks Questions" from Binary Operations→Binary Operations — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Find the points of discontinuity, if any of the following function:
$\text{f(x)}=\begin{cases}\frac{\text{x}^4-16}{\text{x}-2},&\text{if }\text{ x}\neq2\\16,&\text{if }\text{ x}=2\end{cases}$

Prove by Mathematical Induction that$ (A′)^n = (A^n)′,$ where $\text{n}\in\text{N}$ for any square matrix A.

If $\text{A}=\begin{bmatrix}2&-1\\3&2 \end{bmatrix} $ and $\text{B}=\begin{bmatrix}0&4\\-1&7\end{bmatrix} ,$ find $3A^2 - 2B + l$

verify that $\text{y}=\text{-x}-1$ is a solution of the differential equation $(\text{y}-\text{x})\text{dy}-(\text{y}^2-\text{x}^2)\text{dx}=0.$

In a hockey match, both teams $A$ and $B$ scored same number of goals up to the end of the game, so to decide the winner, the referee asked both the captains to throw a die alternately and decided that the team, whose captain gets a six first, will be declared the winner. If the captain of team A was asked to start, find their respective probabilities of winning the match and state whether the decision of the referee was fair or not.

Find the points of discontinuity, if any of the following function:
$\text{f(x)}=\begin{cases}\text{x}^{10}-1,&\text{if }\text{ x}\leq1\\\text{x}^2,&\text{if }\text{ x}>1\end{cases}$

Given $\text{A}=\begin{bmatrix}5 & 0 & 4 \\2 & 3 & 2 \\ 1 & 2 & 1\end{bmatrix},\text{B}^{-1}=\begin{bmatrix}1 & 3 & 3 \\1 & 4 & 3 \\ 1 & 3 & 4\end{bmatrix}.$ Compute $(AB)^{-1}$.

Let f = {(1, -1), (4, -2), (9, -3), (16, 4)} and g = {(-1, -2), (-2, -4), (-3, -6), (4, 8)}. Show that gof is defined while fog is not defined. Also, find gof.

If $\text{A}=\begin{bmatrix}1&-2&0\\ 2&1&3\\ 0&-2&1\end{bmatrix}$, find $A^{-1}$. Using $A^{-1}$, solve the system of linear equations:
$x - 2y = 10, 2x + y + 3z = 8, -2y + z = 7$

Solve the following differential equation:
$\text{xy}\frac{\text{dy}}{\text{dx}}=\text{x}^2-\text{y}^2$