# Math Help - Proving DeMorgan Law?

1. ## Proving DeMorgan Law?

Prove that (A $\cup$ B)c = Ac $\cap$Bc (Demorgan’s Law)
a)
1) Let x be an arbitrary element of (A $\cup$B)c
2) x $\notin$ (A $\cup$B) (By Definition of Complement)
3) x $\notin$ A but x $\in$ Ac (Definition of Union And Definition of Complement)
4) x $\notin$ B but x $\in$ Bc (Definition of Union And Definition of Complement)

5) x $\in$ Ac, x $\in$ Bc <-> x $\in$Ac $\cap$Bc

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

7) Thus (A $\cup$ B)c $\subseteq$ Ac $\cap$Bc

b) And Conversely
1) Let y be an arbitrary element of Ac $\cap$ Bc

2) y $\in$ Ac, y $\in$ Bc (By Definition of intersection)

3) y $\notin$ A, y $\notin$ B (Definition of Complement)

4) y $\notin$ A U B (From b(3))

5) y $\in$ (A U B)c

6) But y is an arbitrary element of Ac $\cap$ Bc

7) Thus Ac $\cap$ Bc $\subseteq$ (A $\cup$ B)c

From a7 and b7 we have

(A $\cup$ B)c = Ac $\cap$ Bc

What is wrong in above statements? any help please?
Thank You.

Note: Solution is also atatched in doc file.