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Question

# Consider the binary operations*: R ΓR β and o: R Γ R β R defined as and a o b = a, &mnForE;a, b β R. Show that * is commutative but not associative, o is associative but not commutative. Further, show that &mnForE;a, b, c β R, a*(b o c) = (a * b) o (a * c). [If it is so, we say that the operation * distributes over the operation o]. Does o distribute over *? Justify your answer.

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Solution

## It is given that *: R ΓR β and o: R Γ R β R isdefined as and a o b = a, &mnForE;a, b β R. For a, b β R, we have: β΄a * b = b * a β΄ The operation * is commutative. It can be observed that, β΄The operation * is not associative. Now, consider the operation o: It can be observed that 1 o 2 = 1 and 2 o 1 = 2. β΄1 o 2 β  2 o 1 (where 1, 2 β R) β΄The operation o is not commutative. Let a, b, c β R. Then, we have: (a o b) o c = a o c = a a o (b o c) = a o b = a β a o b) o c = a o (b o c) β΄ The operation o is associative. Now, let a, b, c β R, then we have: a * (b o c) = a * b = (a * b) o (a * c) = Hence, a * (b o c) = (a * b) o (a * c). Now, 1 o (2 * 3) = (1 o 2) * (1 o 3) = 1 * 1 = β΄1 o (2 * 3) β  (1 o 2) * (1 o 3) (where 1, 2, 3 β R) The operation o does not distribute over *.

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