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Home  >>  CBSE XII  >>  Math  >>  Relations and Functions
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Given a non-empty set \(X\), consider the binary operation \(\ast : P(X) × P(X) \to P(X)\) given by \(A \ast B=A \cap B\; \forall A, \) \( B \;in\; P(X),\) where \(P(X)\) is the power set of \(X\). Show that \(X\) is the identity element for this operation and \(X\) is the only invertible element in \(P(X)\) with respect to the operation \(\ast\).

   
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  • An element X is identify element for a binary operation * if $ A*X=A=X*A$
  • An element A is invertible if then exists B such that $ A*B=X=B*A$
Given $\ast$ defined : $P(X) \times p(X) \to P(X)$ and given by $A*B =A \cap B, A,B \in P(X)$
An element X is identify element for a binary operation * if $ A*X=A=X*A$
$\Rightarrow A \cap X =A=X \cap A \qquad for A \in p(X)$
Therefore we know that $A*X=A=X*A$
Hence X is the identity element .
An element A is invertible if then exists B such that $ A*B=X=B*A$
$A \cap B =X \;and\;B \cap A=X \rightarrow$ This is possible only if $A=B=X$
ie $A *B=X=B*A$ only possible element satisfying the relation is the element X.
answered Feb 28, 2013 by meena.p
edited Mar 20, 2013 by balaji.thirumalai
 

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