Given a non-empty set X, consider the binary operation * : P (X)×P (X)→P (X) given by A * B= A∩B ∀A,B in 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*.
It is given that * : P (X) × P (X) \(\to\) P (X) is defined as A * B = A ∩ B ∀ A, B ∈ P(X)
We know that A ∩ X = A = X ∩ A ∀ A ∈ P (X)
\(\Rightarrow\) A * X = A = X * A ∀ A ∈ P (X)
Thus, X is the identity element for the given binary operation *
Now, an element A ∈ P (X) is invertible if there exists B ∈ P(X)
such that A * B = X = B * A i.e, A∩B= X = B∩A
This case is possible only when A = X = B.
Thus, X is the only invertible element in P (X) with respect to the given operation*.
Hence, the given result is proved.
A school is organizing a debate competition with participants as speakers and judges. $ S = \{S_1, S_2, S_3, S_4\} $ where $ S = \{S_1, S_2, S_3, S_4\} $ represents the set of speakers. The judges are represented by the set: $ J = \{J_1, J_2, J_3\} $ where $ J = \{J_1, J_2, J_3\} $ represents the set of judges. Each speaker can be assigned only one judge. Let $ R $ be a relation from set $ S $ to $ J $ defined as: $ R = \{(x, y) : \text{speaker } x \text{ is judged by judge } y, x \in S, y \in J\} $.
The correct IUPAC name of \([ \text{Pt}(\text{NH}_3)_2\text{Cl}_2 ]^{2+} \) is: