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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?

Rio Alvarado
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dendritic
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    A proper subset is a special kind of subset. Any proper subset is also a subset. Not every subset is a proper subset. In the above example, $C$ is a subset of $A$ and it is also a proper subset of $A$. – Ittay Weiss Sep 01 '15 at 02:27
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    This is also like divisors. The divisors of $8$ are $1,2,4,8$, but the proper divisors of $8$ are $1, 2, 4$. – The Chaz 2.0 Sep 01 '15 at 03:13
  • Generally, in math, $A$ is a "proper ___" of $B$ if $A$ is a ___ of $B$ and $A\ne B$. – Akiva Weinberger Sep 01 '15 at 12:51

3 Answers3

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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 $\emptyset$.

layman
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A proper subset of $S$ is any subset of $S$ that is different from $S$.

Tony Power
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For a subset to be a proper subset there should be one element that is not in the set otherwise they were equal sets.

For ex- B-(1,2,3)

its subsets would be {empty},{1},{2},{3},{1,2},{2,3},{1,3},{1,2,,3}

S any subsets which contains any one of the above elements are proper subsets except the last one.

RajSharma
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