You visited us 1 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard XII

Mathematics

Invertible Element Binary Operation

Given a non-e...

Question

Given a non-empty set X, consider the binary operation : P(X) × P(X) → P(X) given by A B = A ∩ B &mnForE; 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*.

Open in App

Solution

It is given that.

We know that.

Thus, X is the identity element for the given binary operation .

Now, an elementis invertible if there existssuch that

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.

Suggest Corrections

Similar questions

Q. Given a non-empty set

X

, consider the binary operation

* : P (X) \times P (X) \to P (X)

given by

A * B = A \cap B \forall A, B

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

*

Given non-empty set X, consider the binary operation $* : P (X) \times P (X) \to P (X)$ given by $A * B = A \cap B \forall A, B$ 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.

Q. For a non-empty set

X

, if operation

* : P (X) \times P (X) \Rightarrow P (X)

is defined as

A * B = (A - B) \cup (B - A), \forall A, B \in P (X)

then show that empty set

ϕ

is the identity for

*,

and all elements

A

P (X)

are invertible with

A^{- 1} = A

Given a none-empty set $X, l e t * : P (X) \times P (X) \to P (X)$ be defined as $A \times B = (A - B) \cup (B - A), \forall A, B \in P (X)$ . Show that the empty set $ϕ$ is the identity for the operation $*$ and all the elements A of P(X) are invertible with $A^{- 1} = A .$

Q. Let

X

be a non- empty set and

*

be a binary operation on

P (x)

defined by

A * B = A \cap B

for all

A, B \in P (x)

. Find the identity and invertible elements

Join BYJU'S Learning Program

Given a non-empty set X, consider the binary operation *: P(X) × P(X) → P(X) given by A * B = A ∩ B &mnForE; 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*.

Given a non-empty set X, consider the binary operation : P(X) × P(X) → P(X) given by A B = A ∩ B &mnForE; 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*.