Why is the empty set a proper subset of every set?

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  • Thread starter Logical Dog
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In summary: Let us call the set A "A = {apple, banana}" and the set B "B = {apple, banana, cherry}".In summary, the empty set is a subset of all sets, including non-empty sets. This is because a set A is considered a subset of another set B if all elements of A are contained within B, and the empty set has no elements to violate this condition. This also means that the empty set is a proper subset of any non-empty set.
  • #1
Logical Dog
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I know what a proper subset is, but I never understood why every set has the empty set as its subset?

I mean, is the reasoning something primitive like this: if I have x objects, the number of unordered sets of elements I can make are 2^x, including the case where I throw out x objects and don't make anything of them?
 
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  • #2
S is a subset of A iff all elements of S are elements of A. Since the empty set has no elements, this condition is trivially satisfied: the empty set is a subset of all sets. S is a proper subset of A iff S is a subset of A and S is not equal to A. The empty set is therefore a proper subset of any non-empty set.
 
  • #3
TeethWhitener said:
The empty set is therefore a proper subset of any non-empty set.
The "non-empty" is important here. The empty set is not a proper subset of the empty set. It is still a subset.
 
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  • #4
TeethWhitener said:
. Since the empty set has no elements, this condition is trivially satisfied: the empty set is a subset of all sets..

I don't understand why
 
  • #5
mfb said:
The "non-empty" is important here. The empty set is not a proper subset of the empty set. It is still a subset.

A set A is a proper subset of another B set if and only if all elements of A are contained within B, and B has at least one element that is not contained in A.

The ampty set is a proper subset of all sets except itself.

But why?
 
  • #6
Bipolar Demon said:
I don't understand why
Are all elements of ##\emptyset## elements of ##A##?
 
  • #7
TeethWhitener said:
Are all elements of ##\emptyset## elements of ##A##?

There is no intersection, it is a disjoing, but A contains as the empty subset as a subset...so there is an intersection? Now i am confused
 
  • #8
Let's think about it a different way: If, for all ##x \in S##, it's also true that ##x## is an element of ##A##, then ##S \subseteq A##. So now all you have to do is list out all of the ##x \in \emptyset##, and check to make sure that they're all in ##A##.

EDIT: this may require a little knowledge of first-order logic. In symbolic terms, the condition for ##S \subseteq A## is
$$(S \subseteq A) \leftrightarrow \forall x (x \in S \rightarrow x \in A)$$
The rule for material implication is that ##p \rightarrow q## is true when ##p## is false, regardless of what ##q## is. So ##x \in \emptyset## is false for all ##x##, which means the condition $$\forall x (x \in S \rightarrow x \in A)$$
evaluates to
$$\forall x (F \rightarrow x \in A)$$
$$\forall x (T)$$
$$T$$
where ##F## is "false" and ##T## is "true."
 
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  • #9
Let's take a different approach: "Every square number between 5 and 7 is odd". Is this statement correct?

"Every element in ∅ is also element of set A". Is this statement correct?
Same logic.
 
  • #10
All I know about logic is this:

SMl5t4c.png


mISTAKE/typo..UNION Is the OR operator, but yes, this is all I know
 
  • #11
We can try a "proof by contradiction" as well (even though this is a definition, not a theorem). Assume that the empty set isn't a subset of ##A##. Then there exists some element of the empty set that is not an element of ##A##. But this is impossible, because the empty set has no elements. So the empty set has to be a subset of ##A##.
 
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  • #12
Let ##A## be a non-empty set. Can you name me an element of ##\emptyset## that is not in ##A##?
 
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  • #13
micromass said:
Let ##A## be a non-empty set. Can you name me an element of ##\emptyset## that is not in ##A##?
No, and that is why it must be a subset of A.:oldbiggrin:
 
  • #14
Bipolar Demon said:
No, and that is why it must be a subset of A.:oldbiggrin:
Correct.
Bipolar Demon said:
All I know about logic is this:
This is not logic. This is about unions and intersections of sets, which do not matter here.
 
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FAQ: Why is the empty set a proper subset of every set?

1. Why is the empty set a proper subset of every set?

The empty set, also known as the null set, is a set that contains no elements. As a result, it is considered a subset of every set because it does not contain any elements that are not in the larger set. It is also considered a proper subset because it is a subset that is not equal to the larger set.

2. Can a set be a subset of itself?

No, a set cannot be a subset of itself. A set is considered a subset of another set if all elements in the smaller set are also present in the larger set. Since a set cannot contain itself as an element, it cannot be a subset of itself.

3. Why is it important to understand the concept of proper subsets?

Understanding proper subsets is important because it helps us to differentiate between different types of subsets. Proper subsets are subsets that are not equal to the larger set, while improper subsets are subsets that are equal to the larger set. This distinction is useful in various mathematical proofs and sets operations.

4. Can the empty set be a proper subset of the universal set?

Yes, the empty set can be a proper subset of the universal set. The universal set is a set that contains all possible elements, and the empty set is a subset of every set. Therefore, the empty set is also a proper subset of the universal set.

5. How is the empty set represented in mathematical notation?

The empty set is represented by the symbol ∅ or {}. This symbol is used to indicate a set with no elements. It is important to note that this symbol is not the same as the number 0, which represents a numerical value, while the empty set represents the absence of elements in a set.

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