I was answering the questions in my book and came over the question: What is A intersection B complement? I thought the answer would be shading everything EXCEPT the middle part where A and B intersect, but instead when I check my answer from the back, the ONLY thing they shaded was the middle part where A and B intersect. Could anyone please explain this to me?
[Math] A intersection B’
elementary-set-theory
Related Solutions
If $\cal A$ is a collection of sets then we define the two operations:
- The union of $\cal A$ which is $\bigcup{\cal A}=\{x\mid\exists A\in{\cal A}: x\in A\}$. So the elements of the union are all those which are elements of elements of the collection $\cal A$.
- The intersection of $\cal A$ which is $\bigcap{\cal A}=\{x\mid\forall A\in{\cal A}:x\in A\}$. Similarly, the elements of the intersection are those which are elements in all the elements of $\cal A$.
So now we look for $\bigcup F$ and $\bigcap F$. What real numbers are in all the intervals? How can you describe that set nicely? What is the collection of real numbers which appear in some of the sets?
Let me hint you for the first answer, $1$ is clearly in all the intervals and no number smaller than $1$ is in any interval to begin with. For every number $x>1$ we can find an interval $[1,1+\frac1n]$ such that $x$ is not in that interval. So which numbers appear in all the intervals?
To your final paragraph, let me add that $\{1\}$ is in fact the interval $[1,1]$. And it goes to show you that the intersection of non-degenerate intervals doesn't have to be a non-degenerate interval.
For fixed values of $k$, you have $$\bigcup_{n=k}^{\infty} A_n = A_k \cup A_{k+1} \cup A_{k+2} \cup \cdots$$ So $x \in \bigcup\limits_{n = k}^{\infty} A_n$ means that $x \in A_n$ for some $n \ge k$.
Now to say that $x \in \bigcap\limits_{k=1}^{\infty} \bigcup\limits_{n=k}^{\infty} A_k$ is precisely to say that for all $k \ge 1$, there is some $n \ge k$ such that $x \in A_n$.
In plainer words: no matter how large a value of $k$, you give me, I can give you a value of $n$ larger than that $k$ such that $x \in A_n$.
As such, the words you seek are: infinitely many.
You should verify that this is equivalent to the claim that $x \in \bigcap\limits_{k=1}^{\infty} \bigcup\limits_{n=k}^{\infty} A_k$.
Best Answer
$A\cap B^C$ is the definition of $A\setminus B$: everything that is in $A$, but not in $B$. A corresponding Venn-diagram is here:
${}\rlap{\raise3cm{\hskip1.5cm\color{white}{A\setminus B}}}$