Find the cardinality of the following set?

Question

If I have a set A with |A| = n then what would be the cardinality of the following set?

{ | ∈ (A), |X| ≤ 1}

What I understand is that this is the set of all X such that it is part of the power set of A and cardinality is less than or equal to 1. Thus I say the cardinality of this set is 2. Because there are two sets: {}, the empty set and {X} which are part of the power set of A.

Am I understanding correctly?

Answer 1

You seem to be have trouble with set notation. $X$ is a place holder variable that is not any actual set. It merely means "Let $X$ be any element of $P(A)$" which is to say "Let $X$ be any subset of $A$".

So $\{X| X\in P(A); |X| \le 1\} =$ the set of all subsets of $A$ that have a cardinality of at most one.

And the question "what is the cardinality of $\{X| X\in P(A); |X| \le 1\}$" is asking nothing more or less than "how many subsets of $A$ have one or fewer elements".

And the answer to that is obvious. The empty set is a subset with zero elements, and every set with a single element has one element. And any set with more than one element has a cardinality of more than one. As there are $n$ elements in $A$ there are $n$ sets with a single element. Adding the empty set we have the cardinality is $n+1$.

Another way of looking at this is:

If $A = \{a_1,a_2,......, a_n\}$ then $\{X| X\in P(A); |X| \le 1\}=\{\emptyset, \{a_1\},\{a_2\},....., \{a_n\}\}$

Answer 2

You have $1$ subset of $A$ of cardinality $0$ (the empty subset).

And you have $n$ subsets of $A$ of cardinality $1$ (each $\lbrace x \rbrace$ for $x \in A$).

So the set of subsets of cardinality at most $1$ has cardinality $$n+1 $$