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Home  >>  CBSE XII  >>  Math  >>  Relations and Functions
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Let \(\ast\) be the binary operation on \(N\) defined by \(a \ast b=H.C.F.\,of\,a\,\;and\;\,b\). Is \(\ast\) commutative? Is \(\ast\) associative? Does there exist identity for this binary operation on \(N\)?

$\begin{array}{1 1} \text{Not Commutative, Not Associative, Identity Exists} \\ \text{Not Commutative, Not Associative, Identity Does not Exist} \\ \text{Commutative, Associative, Identity Exists} \\ \text{Commutative, Associative, Identity Does not Exist} \end{array} $

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Toolbox:
  • An operation $\ast$ on $A$ is commutative if $a\ast b = b \ast a\; \forall \; a, b \in A$
  • An operation $\ast$ on $A$ is associative if $a\ast ( b \ast c) = (a \ast b) \ast c\; \forall \; a, b, c \in A$
  • An element $e \in N $ is an identify element for operation * if $a*e=e*a$ for all $a \in N$
  • The highest common factor (HCF), of two or more non-zero integers, is the largest positive integer that divides the numbers without a remainder.
Given that the binary operation on N given by $a \ast b = $ HCF $(a,b)$:
$\textbf {Step 1: Checking if the operation is commutative}$:
We know $ HCF $(a,b) = $HCF $(b,a)$.
$\Rightarrow a*b=b*a$ for all $a,b \in N \rightarrow \ast $ is commutative.
$\textbf {Step 2: Checking if the operation is associative}$:
For an operation $\ast$ to be associative $a\ast ( b \ast c) = (a \ast b) \ast c\; \forall \; a, b, c \in N$
$(a*b)*c = $ HCF $(a,b) \ast c =$ HCF $(a,b,c)$
$ a*(b*c)= a*($ HCF $(b,c)) =$ HCF $(a,b,c)$
$\Rightarrow (a*b)*c=a*(b*c) \forall a,b,c \in N \rightarrow \ast $ is associative.
$\textbf {Step 2: Checking if the operation has an identity}$:
We know that the element $e \in N $ is an identify element for operation * if $a*e=e*a$ for all $a \in N$
We can clearly see that this is NOT true for all $a \in N$ through a small example:
Let $a=5$, $ 5 \ast 1 = 1$ and so is $6 \ast 1 = 1$.
So, this relation is not true for al $a \in N$. Therefore $\ast$ does not have an identity element in $N$.
answered Feb 27, 2013 by meena.p
edited Mar 19, 2013 by balaji.thirumalai
 

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