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Question
Let S be the set of all real numbers except −1 and let '*' be an operation defined by a * b = a + b + ab for all a, b ∈ S. Determine whether '*' is a binary operation on S. If yes, check its commutativity and associativity. Also, solve the equation (2 * x) * 3 = 7.
Solution
Checking for binary operation:
\[\text{Let }a, b \in S . \text{Then}, \]
\[a, b \in \text{R and a} \neq - 1, b \neq - 1\]
\[a * b = a + b + ab\]
\[\text{We need to prove thata} + b + ab \in S . \left[ \text{For this we have to prove thata} + b + ab \in \text{ R and a } + b + ab \neq - 1 \right]\]
\[\text{Since a, b} \in R, a + b + ab \in R, \text{let us assume thata} + b + ab = - 1 . \]
\[a + b + ab + 1 = 0\]
\[a + ab + b + 1 = 0\]
\[a\left( 1 + b \right) + 1\left( 1 + b \right) = 0\]
\[\left( a + 1 \right)\left( b + 1 \right) = 0\]
\[a = - 1, b = - 1 \left[ \text{which is false} \right]\]
\[\text{Hence},a + b + ab \neq - 1\]
\[\text{Therefore},\]
\[a + b + ab \in S\]
Thus, * is a binary operation on S.
Commutativity:
\[\text{Let }a, b \in S . \text{Then}, \]
\[a * b = a + b + ab\]
\[ = b + a + ba\]
\[ = b * a \]
\[\text{Therefore},\]
\[a * b = b * a, \forall a, b \in S\]
Thus, * is commutative on N.
Associativity :
\[\text{Let a}, b, c \in S\]
\[a * \left( b * c \right) = a * \left( b + c + bc \right)\]
\[ = a + b + c + bc + a\left( b + c + bc \right)\]
\[ = a + b + c + bc + ab + ac + abc\]
\[\left( a * b \right) * c = \left( a + b + ab \right) * c\]
\[ = a + b + ab + c + \left( a + b + ab \right)c\]
\[ = a + b + ab + c + ac + bc + abc\]
\[\text{Therefore},\]
\[a * \left( b * c \right) = \left( a * b \right) * c, \forall a, b, c \in S\]
Thus, * is associative on S.
Now,
\[\text{Given}:\hspace{0.167em}\left( 2 * x \right) * 3 = 7\]
\[ \Rightarrow \left( 2 + x + 2x \right) * 3 = 7\]
\[ \Rightarrow \left( 2 + 3x \right) * 3 = 7\]
\[ \Rightarrow 2 + 3x + 3 + \left( 2 + 3x \right)3 = 7\]
\[ \Rightarrow 5 + 3x + 6 + 9x = 7\]
\[ \Rightarrow 12x + 11 = 7\]
\[ \Rightarrow 12x = - 4\]
\[ \Rightarrow x = \frac{- 4}{12}\]
\[ \Rightarrow x = \frac{- 1}{3}\]
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