Prove that (A B)c = Ac Bc (Demorgan’s Law)

a)

1) Let x be an arbitrary element of (A B)c

2) x (A B) (By Definition of Complement)

3) x A but x Ac (Definition of Union And Definition of Complement)

4) x B but x Bc (Definition of Union And Definition of Complement)

5) x Ac, x Bc <-> x Ac Bc

6) But x is an arbitrary element of (AUB)c

7) Thus (A B)c Ac Bc

b) And Conversely

1) Let y be an arbitrary element of Ac Bc

2) y Ac, y Bc (By Definition of intersection)

3) y A, y B (Definition of Complement)

4) y A U B (From b(3))

5) y (A U B)c

6) But y is an arbitrary element of Ac Bc

7) Thus Ac Bc (A B)c

From a7 and b7 we have

(A B)c = Ac Bc

What is wrong in above statements? any help please?

Thank You.

Note: Solution is also atatched in doc file.