Question

If A ={a} ,B= {b} , Prove that P(A) U P(B)not equal to P(AUB)​

Answer 1

Given:

• A = {a}
• B = {b}

To prove:

• P(A) U P(B) ≠ P(A U B)

Solution:

~Finding power sets ::

>> Given set A = {a}

Subsets are the sub part of a set. So the subsets of A will be {ø} and {a}.

Power set of a set is the set of all its subset. So the power set of A will be P(A) = { {ø} , {a} }

>> Given set B {b}

Subsets of set B = {b} and {ø}.

It's power set will be { {b} , {ø} }

~Finding union ::

Union of two set is the set containing elements of both the sets. In simple words Union is the addition of two sets left behind the common elements (since an element can't be repeated in a set).

Union of power set of A and B is ::

→ P(A) U P(B) = { {ø}, {a} , {b} } ...(i)

Power set of union of A and B::

→ A U B = { a , b }

Subsets of (A U B) are {a} , {b} , {ø} and {a,b}

Power set of (A U B), P(A U B) = {{a}, {b}, {ø}, {a,b} } ..(ii)

Clearly from (i) and (ii) ::

→ P(A) U P(B) ≠ P(A U B)