Proof of Set Theory: A \subseteq B implies Bc \subseteq Ac

[cmajor47]

Homework Statement

For all sets A and B, if A $$\subseteq$$ B then B^c $$\subseteq$$ A^c.

Homework Equations

The Attempt at a Solution

Proof: Suppose A and B are sets and A $$\subseteq$$ B.
Let x $$\in$$ B^c
By definition of complement, if x $$\in$$ B^c then x $$\notin$$ B
Since x $$\notin$$ B, x $$\notin$$ A
Since x $$\notin$$ A, x $$\in$$ A^c by definition of complement
Therefore if A $$\subseteq$$ B then B^c $$\subseteq$$ A^c.

I just want to make sure that this proof is correct and that there are no mistakes. Thanks!

 

[Dick]

It's fine. You might want to enhance it's proofiness by stating the reason why x not in B implies x not in A as you gave a reason for the other lines.