How many elements are present in the subset of null set ?
How many elements are present in the subset of a null set?
This is one the question that appeared in my math exam.
Definition $1.1$ - Subset: A set $A$ is a subset of set $B$ if all the elements of $A$ are also elements of $B$
Definition $1.2$ - Null set or Void set or Empty set: If is a set containing no elements
Definition $1.3$ - Power set: It is the set of all possible subsets of a given set
Theorem $1.1$: Every set is a subset of itself
Theorem $1.2$: Null set is a subset of every set
I think the answer to this question is $0$ because,
$$\mathbf{No.\ of\ subsets} = 2^m$$So, the number of subsets of a null set (denoted by $\emptyset$) which contains $0$ elements would be $2^0 = 1$ and that subset will be the null set $\emptyset$ itself. Hence, the number of elements in $0$.
But my math teacher told me that the answer is $1$. And her reasoning is as follows, she stated the same that the number of subset of a null set will be $1$ and she represented subset of null set as {$\emptyset$}. So she told the answer to be $1$ as the null set acted as an element in here.
I don't know which of the answers - $0$ or $1$ is correct. There is a debate among me and my teacher about the answers. So, you answers with explanation helps me. Could someone let me know . . .
1 answer
The empty set (or the null set) has no elements. Notation: {}
The empty set has a single subset, which is the empty set. Any other set has at least one element in it, so it can not be a subset of the empty set. Hence the empty set is the only subset of itself.
The power set of the empty set consists of all of its subsets, that is, it consists of the empty set. Notation: {{}}. (A set which has a single element which is the empty set. The inner curly brackets are the empty set, while the outer ones are the set that contains the empty set.) The power set of the empty set has a single element.
My guess is that one of you thought about the number of elements of the empty set (0) and the other thought about the number of elements of the power set of the empty set (1). The number of elements of the only subset of the empty set is also zero, but it is a bit of a strange question.
To add more complications: The power set of the empty set has two subsets; the empty set is a subset and the power set itself is its own subset, much like any set is its own subset. One of these has no elements and the other has a single element.
Especially if you are talking, it is really easy to stumble here. I suggest drawings.
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