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Is \(\ast\) defined on the set \(\{1,2,3,4,5\}\) by \(a\ast b=L.C.M\).of \(a\;and\;b\) a binary operation?

$\begin{array}{1 1} \text{at is a binary operation} \\ \text{at is not a binary operation } \end{array}$

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  • The lowest common multiple of two integers a and b, usually denoted by LCM(a, b), is the smallest positive integer that is divisible by both a and b.
  • A binary operation $\ast$ on a set $A$ is a function $\ast$ from $A \times A$ to $ A$. Therefore, if $a,b \in A \Rightarrow a \ast b \in A\; \forall\; a,b, \in A$
Given $\ast$ on set $A: \{1,2,3,4,5\}$ is defined by $a \ast b =$ LCM $(a,b)$:
Consider $a=1, b=2 \rightarrow 1 \ast 2 = $ LCM $(1,2) = 2$ and $2 \in A$.
However, if we consider another pair of elements $a=2, b=3 \rightarrow 2 \ast 3 =$ LCM$(2,3) = 6$ and $6 \not \in A$
Therefore, $\ast$ is not a binary operation.
answered Feb 27, 2013 by meena.p
edited Mar 19, 2013 by balaji.thirumalai

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