(a) To prove that B\A = (AB)* using only the above equations and equivalences, we need to show that B\A is equivalent to (AB)*.
First, we will show that B\A is a subset of (AB)*.
Let x be an element of B\A. Then, x is in B and x is not in A. Therefore, x is in AB and not in A. This means that x is in (AB)*. Thus, we have shown that B\A is a subset of (AB)*.
Next, we will show that (AB)* is a subset of B\A.
Let x be an element of (AB)*. Then, x is in AB or x is not in AB.
If x is in AB, then x is in B and x is in A. Therefore, x is not in B\A.
If x is not in AB, then x is not in A or x is not in B. Therefore, x is not in A and x is in B. This means that x is in B\A.
Thus, we have shown that (AB)* is a subset of B\A.
Since we have shown that B\A is a subset of (AB)* and (AB)* is a subset of B\A, we can conclude that B\A = (AB)*.
(b) To prove that B* A* = (AB)*, we need to show that B* A* is a subset of (AB)* and (AB)* is a subset of B* A*.
First, we will show that B* A* is a subset of (AB)*.
Let x be an element of B* A*. Then, x is in (B*) and x is in (A*).
If x is in B*, then x is in B or x is not in B.
If x is in A*, then x is in A or x is not in A.
If x is in B and x is in A, then x is in AB.
If x is not in B and x is not in A, then x is not in AB.
If x is in B and x is not in A, then x is in B\A.
If x is not in B and x is in A, then x is in A\B.
Therefore, we have shown that x is in (AB)*.
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