I'm a bit confused on the wording here..
For example:
$$A = \{c, d, f, g\}$$
$$C = \{d, g\}$$
Is $C$ "subset" of $A$? Obviously, yes.
But.. the proper subset states that:
If $C$ and $A$ are any sets, then $C$ is a proper subset of $A$ if and only if $C$ is a subset of $A$, BUT there exists some elements of $A$ that is NOT in $C$.
So, would $C$ "subset" of $A$ be FALSE? Instead, the correct answer is $C$ is "proper subset" of $A$? Because there are some elements of $A$ that are not in $C$.
Could anyone clarify this?
Best Answer
Subset is more general than proper subset (like green is more general than dark green -- everything that is dark green is green, but not everything that is green is dark green).
A proper subset of $A$ is a subset of $A$ that is not equal to $A$. So if $A = \{1, 2\}$, then the subsets of $A$ are $\emptyset$, $\{1\}$, $\{2\}$, and $\{1,2\}$.
The first three are proper subsets of $A$ since they are subsets of $A$, but they aren't equal to $A$. The other subset of $A$, $\{1,2\}$, is not a proper subset of $A$, since it equals $A$.
So basically, for any non-empty set $A$, if you think of all of its subsets, then all but one of them is proper. The only one that isn't proper is $A$ itself.
Note, if $A = \emptyset$, then its only subset is itself, so that shows you the empty set has no proper subsets, since it only has one subset, which is $\emptyset$.