This problem is from Discrete Mathematics and its Applications
My question is on 9b. I know that the sign represents an element is a member of.
(from book)
I know that the O with a slash across it is the empty set which "is a special set that has no elements".
From http://mathcentral.uregina.ca/QQ/database/QQ.09.06/narayana1.html, I got that the empty set is a subset of all sets, meaning that every member of the empty set(nothing) is also a member of any other set.
Based on all of this, for 9b, would {0} contain the empty set because it fundamentally has the elements that consist of the empty set(nothing) or does it physically have to
have the empty set?
Best Answer
When $X$ and $Y$ are two sets, we say that $X\subset Y$ if every element of $X$ is contained in $Y$.
With this definition, you see that $\emptyset \subset Y$ for any set $Y$. Indeed, there is no element in $\emptyset$, so every element of $\emptyset$ is contained in $Y$ (trivially true as there is nothing to check).
However, if you want to write $\emptyset \in Y$, this means that there is one element of $Y$ which is a set and that this set is the empty set. When $Y=\{0\}$, you have only one element in $Y$, and this one is not a set, it is a number, which is $0$. Hence, $\emptyset\notin \{0\}$.
Both statements $9a$ and $9b$ are false.