Prove that $A-(A-B)=A \cap B$ by use set properties [closed]
How do you use set properties to prove that $A-(A-B)=A \cap B$ ?
Solution 1:
You have that by definition $X-Y=X\cap Y^c$. Thus $$A-(A-B)=A\cap(A-B)^c=A\cap(A\cap B^c)^c\\=A\cap(A^c\cup B)=(A\cap A^c)\cup (A\cap B)=\emptyset \cup (A\cap B)=A\cap B.$$ Do you recognize the "set properties" that I used?