0
$\begingroup$

Can you help me out? It's been quite a pain in the backseat lately.

Question: Let $A$ be any set with $|A|=m$. Find $|\{x\in P(A):|x|\le 1\}|$ (where $P(A)$ is the power set of $A$).

Hope you will find time to answer this!

  • 0
    How can an element of the power set be compared to 1 ? – 2017-02-09
  • 0
    Think about what $|x| \le 1$ means. What does $|x| = 0$ mean? What does $|x| = 1$ mean? If $x \in P(A)$ then $x$ is a set. $x$ is a subset of $A$. If $|x|=1$ what elements of A can belong in $x$. How many possible different subsets are there so that $|x| = 1$? That $|x| = 0$? – 2017-02-09

2 Answers 2

2

How many distinct subsets with one element can you find in a set with $m$ elements? How many with less than one element (that is, no element)?

For the first subquestion, try first with $A=\{1,2,3\}$ and make a conjecture. The second subquestion should be easy.

  • 0
    Thanks for the answer! :). I appreciate that you took time to give an answer! May you continue to share your knowledge. – 2017-02-09
0

$\{x \in P(A): |x| \le 1\}=$

$\{x \subset A| x$ has $0$ or $1$ elements$\}=$

$\{x|x = \{a\}$ for some $a \in A$ or $x = \emptyset\}$

Meanwhile....

$|A| = m$ means $A = \{a_1, a_2,......, a_m\}$ for some distinct $a_i$ elements. There are $m$ of these elements.

So $\{x|x = \{a\}$ for some $a \in A$ or $x = \emptyset\}=$

$\{\emptyset, \{a_1\}, \{a_2\},......,\{a_m\}\}$

What is $|\{\emptyset, \{a_1\}, \{a_2\},......,\{a_m\}\}|$?

  • 0
    Thanks for the answer! :). I appreciate that you took time to give an answer! May you continue to share your knowledge. – 2017-02-09