Is every element in a power set a sub set?

I have understood so far that an element cannot be a sub set of itself. If A = {1,2,{3}} and {3} is not a sub set of A. But in my textbook it has been given that every element in a power set is a subset. So isn't there a contradiction?

• Every element of the power-set of a set $A$ is a subset of the set $A$. This is so simply because the power set is defined as the set of all subsets. – Mauro ALLEGRANZA Apr 15 at 12:10
• Okay that makes sense – user662650 Apr 15 at 12:10
• And YES : $\{ 3 \}$ is not a subset of $A$. It is an element of $A$. – Mauro ALLEGRANZA Apr 15 at 12:11
• It is worth pointing out that although $\{3\}$ is not a subset of $A$, $\{\{3\}\}$ is. Note the subtle difference between the two. $\{3\}$ is a set with one element, that element being the number $3$. On the other hand $\{\{3\}\}$ is also a set with one element, but in this case that element is the set which contains $3$. – JMoravitz Apr 15 at 12:17
• It's also just not true, either, that an element $x\in A$ can't also be a subset of $A$. Consider the usual implementation of the natural number 2, $\{\varnothing,\{\varnothing\}\}$; both members of this set are also subsets of this set. – Malice Vidrine Apr 16 at 12:22

3 Answers

Every element of a power set is a subset of the set you formed the powerset for, not of the powerset itself. $$pow(A) := \{X: X \subseteq A\}$$, so by definition, every element of $$pow(A)$$ is a subset of $$A$$, but no element of $$pow(A)$$ is a subset of $$pow(A)$$, and no element of $$A$$ is a subset of $$A$$.
As you correctly noted, $$\{3\}$$ - a set with one element, namely the element $$3$$ - is not a subset, but an element of $$\{A\}$$. But $$\{\{3\}\}$$ - a set with one element, namely the element $$\{3\}$$ - is an element of the powerset of $$A$$ and therefore a subset of $$A$$.

Maybe you should avoid speaking of " an element" or of " a subset" in an absolute sense, as if there were categories of things, some being once and for all and by nature elements, or once and for all and by nature subsets.

In fact the terms " element" and "subset" are relational : element OF a given set, subset OF a given set. One cannot be simply " a brother" , one has to be a brother OF someone...same thing for sets.

For example, take take the object {4}. This object is a set. Relatively to the set

{{4}, { 5,6} }

our object {4} is an element. So we can say that :

the set {4} belongs to the set {{4}, { 5,6} }. Notice that the set {{4}, { 5,6} } has 2 elements, and each of these two elements is a set.

Now, relatively to the set { 4,5,6} ( which has 3 elements), our object {4} is not an element. There is absolutely no contradiction here, if you keep in mind that "element" is a relative term: "being an element OF a given set" does not contradict "not being an element of another set" ( in the same manner as " Peter is a brother of John" does not contradict " Peter is not a brother of Alice").

So what is the object {4} relatively to the set { 4,5,6}? The proper relation here is the inclusion relation : {4} is included in the set { 4,5,6}, it is a subset of { 4,5,6} ( some say also " a part of" { 4,5,6} ).

How to explain that {4} is included in the set { 4,5,6}? Remember that " a set X is included in a set Y just in case all the elements of X are also elements of Y". Now, can you see an element of the one-element set {4} that is not also an element of the three-elements set { 4,5,6}? If you answer "no", you know that the set {4} "passes the test" of inclusion , and is therefore included in { 4,5,6}.

Since the set {4} is a subset of { 4,5,6} ( is included in that set), the set {4} is an element of the power set of the set { 4,5,6}. That comes from the definition of a power set. By definition, the power set of { 4,5,6} is the collection of the subsets of the set { 4,5,6}.

Being given that {4} is a subset of { 4,5,6}, it follows that the set {4}is an element of the power set of { 4,5,6}. And that is logical : every subset of a set S is an element of the collection of the subsets of S, and the power set of S is just this collection.

Remark. DEFINITION : the power set of a set S is the set that has as elements all the subsets of S.

Now let us consider some general " laws" regarding these relations ( element of, subset of)

(1) No set is an element of itself. ( But nothing prevents a set from being an element of another set).

(2) Every set is a subset of itself, every set is included in itself. ( Take any set, say {a, b,c}. Can you see an element of the set {a, b,c} that is not an element of the set {a, b,c}? Of course not! So the set {a, b,c} passes the test of inclusion, relatively to itself: the set {a, b,c} is a subset of the set {a, b,c}, that is to say, of itself. I said subset, not member!)

(3) Every set is an element of its power set .

Explanation :

(a) If a set X is a subset of a set Y, then X is an element of the power set of Y.

(b) But, the set Y is itself a subset of Y ( itself), because of law (2) above.

(c) Therefore, the set Y is an element of its own power set.

A set $$B$$ is a subset of $$A \iff (x \in B \Rightarrow x \in A).$$ For your example take $$B = \{ 3\}$$ then $$B$$ is a subset of $$A$$ if and only if $$3 \in A$$, but this is not the case since $$A$$ only contains $$1,2,\{ 3\}$$ all of which are not $$3$$. Therefore $$B$$ is not a subset of $$A$$.

The power set of a set $$A$$ is by definition the collection of all subsets of $$A$$ so any subset of $$A$$ is by definition in $$A$$'s power set.