Let $$ be an operation such that $a b =$ LCM of $a$ and $b$ defined on the set $A = {1, 2, 3, 4, 5}$ . Is $*$ a binary operation? Justify your answer.

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Solution

Given : $A = {1, 2, 3, 4, 5}$ and $a * b =$ LCM of $a$ and $b$
To check $$ is binary i.e. $a b = b * a$ for any $a, b \in A$
LCM of $2$ and $4$ is $4$ . So, for $a, b = 2 or 4$ , $a * b = 4 = b * a$
Otherwise, LCM of $a$ and $b = a \times b =$ LCM of $b$ and $a$
For example, LCM of $1$ and $2 = 2$ which same as LCM of $2$ and $1$
Thus, $a * b = b * a$ for any $a, b \in A$
Hence, $*$ is binary.

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Let ∗ be an operation such that a∗b=LCM of a and b defined on the set A={1,2,3,4,5}. Is ∗ a binary operation? Justify your answer.

Let $$ be an operation such that $a b =$ LCM of $a$ and $b$ defined on the set $A = {1, 2, 3, 4, 5}$ . Is $*$ a binary operation? Justify your answer.