Prove that for any two sets A;B , we have (A n B)^c = A^c u B^c
I think im supposed to start expansion with the de morgan laws my teacher said I was doing it wrong I'm pretty sure this is the correct approach but i dont know how to continue
Prove that for any two sets A;B , we have (A n B)^c = A^c u B^c
I think im supposed to start expansion with the de morgan laws my teacher said I was doing it wrong I'm pretty sure this is the correct approach but i dont know how to continue
Lol usually if your lecturer tells you it's wrong it's wrong...
Consider an element $\displaystyle x \in (A \cap B)^c$.
Then x is NOT contained in the intersection of A and B.
Hence x is NOT contained in A or B.
Hence x IS contained in $\displaystyle A^c$ and also $\displaystyle B^c$.
So $\displaystyle x \in A^c \cup B^c$.
Then to prove the reverse...
Well, you try it.
It is the correct approach.
Now, after stating $\displaystyle x \in A^c \cup B^c \Rightarrow x \in A^c$ OR $\displaystyle x \in B^c$, separate to 2 cases:
1) If $\displaystyle x \in A^c$, is x in $\displaystyle (A \cap B)^c$?
2) If $\displaystyle x \in B^c$, is x in $\displaystyle (A \cap B)^c$?
Try to formally write this, it usually helps..