Question

Abstract Algebra

Answer 1

Answer:

1.No, because inverse does not exist.

2.No, because inverse does not exist.

Step-by-step explanation:

We are given that X be  a set and let P(X) be the power set of X.

a. We have to tell P(X) with binary operation

A*B=$A\cap B$ form  a group.

Suppose, x={1,2}

P(X)={$\phi$,{1},{2},{1,2}}

1.Closure  property:$A\cap B\in P(X)$

{1}$\cap$ {2}=$\phi \in P(X)$

It is satisfied for all $A,B\in P(X)$

2.Associative property:$(A\cap B)\cap C=A\cap (B\cap C)$

If A={1},B={2},C={1,2}

$A\cap(B\cap C)$={1}$\cap$({2}$\cap${1,2})={1}$\cap$ {2}=$\phi$

$(A\cap B)\cap C$=({1}$\cap${2})$\cap${1,2}=$\phi\cap${1,2}=$\phi$

Hence, P(X) satisfied the associative property.

3.Identity :$A\cap B=A$ Where B is identity element of P(X)

$A\cap X=A$

It is satisfied for every element A in P(X).

Hence, X is identity element in  P(X)

4.Inverse :$A\cap B=X$ Where B is an inverse element of A in P(x)

It can not be possible for every element that satisfied $A\cap B=X$

Hence, inverse does not exist.

Therefore, P(X) is not a  group w.r.t to given binary operation.

2.We have to tell P(X) with the binary operation

A*B=$A\cup B$ form a group

Similarly,

For set X={1,2}

P(X)={$\phi$,{1},{2},{1,2}}

1.Closure property:If A and B are belongs to P(X) then $A\cup B\in P(X)$ for all A and B belongs to P(X).

2.Associative property:$A\cup (B\cup C)=(A\cup B)\cup C$

If A={1},B={2},C{1,2}

$A\cup B$={1}$\cup${2}={1,2}

$(A\cup B)\cup C$={1,2}$\cup${1,2}={1,2}

$B\cup C$={2}$\cup${1,2}={1,2}

$A\cup (B\cup C)$={1}$\cup${1,2}={1,2}

Hence, P(X) satisfied the associative property.

3.Identity :$A\cup B=A$ Where B is identity element of P(X)

Only $\phi$ is that element for every A in P(X) that satisfied $A\cup B=A$

Hence, $\phi$ is identity element of P(X) w.r.t union.

4.Inverse element :

$A\cup B=\phi$ where B is  an inverse element of A in P(X)

It is not possible for every element that satisfied the property.

Hence, inverse does not exist for each element in P(X).

Therefore, P(X) is not  a group w.r.t binary operation.