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Home » Let be a nonempty set and be a binary operation on , the power set of , defined by E for all (i) Find the identity element in . (ii) Show that is the only invertible element in .
Let X be a nonempty set and * be a binary operation on P(X), the power set of X, defined by A * E A \cap B for all A, B \in P(X) (i) Find the identity element in \mathrm{P}(\mathrm{X}). (ii) Show that X is the only invertible element in P(X).
Binary Operations | CBSE | Class 12 | Exercise 3A | Maths | RS Aggarwal
Let X be a nonempty set and * be a binary operation on P(X), the power set of X, defined by A * E A \cap B for all A, B \in P(X) (i) Find the identity element in \mathrm{P}(\mathrm{X}). (ii) Show that X is the only invertible element in P(X).

e is the identity of * if e^{*} a=a

\mathrm{A}

From the Venn diagram,

\begin{array}{l} A * X=A \cap X=A \\ X * A=X \cap A=A \end{array}

\Rightarrow \mathrm{X} is the identity element for binary operation *

Let B be the invertible element

\begin{array}{l} \Rightarrow \mathrm{A}^{*} \mathrm{~B}=\mathrm{X} \\ \Rightarrow \mathrm{A} \cap \mathrm{B}=\mathrm{X} \end{array}

This is only possible if A=B=X

Thus X is the only invertible element in P(X)

i

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