(i) Since,
![\[\begin{array}{*{35}{l}} a\text{ }*\text{ }b\text{ }=\text{ }1.c.m.\text{ }\left( a,\text{ }b \right) \\ 2\text{ }*\text{ }4\text{ }=\text{ }l.c.m.\text{ }\left( 2,\text{ }4 \right) \\ =\text{ }4 \\ 3\text{ }*\text{ }5\text{ }=\text{ }l.c.m.\text{ }\left( 3,\text{ }5 \right) \\ =\text{ }15 \\ 1\text{ }*\text{ }6\text{ }=\text{ }l.c.m.\text{ }\left( 1,\text{ }6 \right) \\ =\text{ }6 \\ \end{array}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-a10457d782f37a3540e39a626c714b6b_l3.png "Rendered by QuickLaTeX.com")
(ii)\ to prove : commutativity of *
![\[\begin{array}{*{35}{l}} Let\text{ }a,\text{ }b\text{ }\in \text{ }N \\ a\text{ }*\text{ }b\text{ }=\text{ }l.c.m\text{ }\left( a,\text{ }b \right) \\ =\text{ }l.c.m\text{ }\left( b,\text{ }a \right) \\ =\text{ }b\text{ }*\text{ }a \\ =>a\text{ }*\text{ }b\text{ }=\text{ }b\text{ }*\text{ }a\text{ }\forall \text{ }a,\text{ }b\text{ }\in \text{ }N \\ \end{array}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-d8cac5159441deabf0f3bc45abcd6a88_l3.png "Rendered by QuickLaTeX.com")
.
Thus * is commutative on N.
Now we have to prove associativity of *
![\[\begin{array}{*{35}{l}} Let\text{ }a,\text{ }b,\text{ }c\text{ }\in \text{ }N \\ a\text{ }*\text{ }\left( b\text{ }*\text{ }c\text{ } \right)\text{ }=\text{ }a\text{ }*\text{ }l.c.m.\text{ }\left( b,\text{ }c \right) \\ =\text{ }l.c.m.\text{ }\left( a,\text{ }\left( b,\text{ }c \right) \right) \\ =\text{ }l.c.m\text{ }\left( a,\text{ }b,\text{ }c \right) \\ \left( a\text{ }*\text{ }b \right)\text{ }*\text{ }c\text{ }=\text{ }l.c.m.\text{ }\left( a,\text{ }b \right)\text{ }*\text{ }c \\ =\text{ }l.c.m.\text{ }\left( \left( a,\text{ }b \right),\text{ }c \right) \\ =\text{ }l.c.m.\text{ }\left( a,\text{ }b,\text{ }c \right) \\ \end{array}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-2eaa5c5d10517f6cbc394814f6360d3b_l3.png "Rendered by QuickLaTeX.com")
Therefore
![\[(a\text{ }*\text{ }\left( b\text{ }*\text{ }c \right)\text{ }=\text{ }\left( a\text{ }*\text{ }b \right)\text{ }*\text{ }c,\text{ }\forall \text{ }a,\text{ }b\text{ },\text{ }c\text{ }\in \text{ }N\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-0c0597fed56347120b11267d2c98da17_l3.png "Rendered by QuickLaTeX.com")
Thus, * is associative on N.
Let ‘*’ be a binary operation on N defined by a * b = l.c.m. (a, b) for all a, b ∈ N (i) Find 2 * 4, 3 * 5, 1 * 6. (ii) Check the commutativity and associativity of ‘*’ on N.
Let ‘*’ be a binary operation on N defined by a * b = l.c.m. (a, b) for all a, b ∈ N (i) Find 2 * 4, 3 * 5, 1 * 6. (ii) Check the commutativity and associativity of ‘*’ on N.