# More basic topology (couple questions)

• May 4th 2011, 08:50 AM
MathSucker
More basic topology (couple questions)
1) I have to prove: \$\displaystyle (A \cup B) - (A \cap B) = (A-B) \cup (B-A)\$

I know some people get annoyed by the "-" notation, but that is what my prof uses, so...

Here is my work:

\$\displaystyle (A \cup B) - (A \cap B) = (A \cup B) \cap (A \cap B)^c\$

\$\displaystyle =(A \cup B) \cap (A^c \cup B^c)\$

\$\displaystyle =(A \cup B) \cup (B \cap A^c)\$

\$\displaystyle =(A-B) \cup (B-A)\$

2) Let \$\displaystyle X = \{ a,b,c,d \}\$

and let \$\displaystyle S = \{ \O , \{ a,b \} , \{ b,c \} , \{ c,d \} , \{ a,d \} , \{ a,b,c,d \} \}\$

I have to find the smallest topology on \$\displaystyle X\$ that contains \$\displaystyle S\$.

So, using the axioms for a topological space:

\$\displaystyle \{ \{ S, \{ a \} , \{ b \} , \{ c \} , \{ d \} , \{ b,d \} , \{ a,c \} , \{ a,b,c \} , \{ a,b,d \} , \{ b,c,d \} , \{ a,c,d \} \}\$

Correct?
• May 4th 2011, 09:08 AM
Plato
You missed an important equality.
\$\displaystyle (A \cup B) \cap (A^c \cup B^c)=(A\cap B^c)\cup(B\cap A^c)\$.
• May 4th 2011, 09:19 AM
MathSucker
Oh right, so it should have been:

\$\displaystyle (A \cup B) - (A \cap B) = (A \cup B) \cap (A \cap B)^c\$

\$\displaystyle =(A \cup B) \cap (A^c \cup B^c)\$

\$\displaystyle =(A \cap B^c) \cup (B \cap A^c)\$

\$\displaystyle =(A-B) \cup (B-A)\$