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Statements about the set 2^M

mathmari

Well-known member
MHB Site Helper
Apr 14, 2013
4,008
Hey!! :eek:

  1. Let $M:=\{7,4,0,3\}$. Determine $2^M$.
  2. Prove or disprove $2^{A\times B}=\{A'\times B'\mid A'\subseteq A, B'\subseteq B\}$.
  3. Let $a\neq b\in \mathbb{R}$ and $M:=2^{\{a,b\}}$. Determine $2^M$.
  4. Is there a set $M$, such that $2^M=\emptyset$ ?

First of all how is $2^M$ defined? Is this the powerset? (Wondering)
 

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,687

mathmari

Well-known member
MHB Site Helper
Apr 14, 2013
4,008
Yep.
So $2^{\{a,b\}}=\{\varnothing,\{a\},\{b\},\{a,b\}\}$. (Thinking)
So we have the following:
  1. $2^M=\{\emptyset, \{7\}, \{4\}, \{0\}, \{3\}, \{7,4\}, \{7,0\}, \{7,3\}, \{4,0\}, \{4,3\}, \{0,3\},\{7,4,0\}, \{7,4,3\}, \{7,0,3\}, \{4,0,3\}, \{7,4,0,3\}$
  2. $2^{A\times B}=\text{ set of every subset of } A\times B=\{(a,b)\mid a\in A, b\in B\}$.

    So it is of the form $2^{A\times B}=\{\emptyset, \{(a_i, b_i)\}, \{(a_i, b_i), (a_j, b_j)\}, \cdots \}$, or not?

    This is not of the form $\{A'\times B'\mid A'\subseteq A, B'\subseteq B\}$, is it? (Wondering)
  3. We have that $M=\{\emptyset, \{a\}, \{b\}, \{a,b\}\}$.

    The $2^M=\{\emptyset, \{\emptyset\}, \{\{a\}\},\{\{b\}\}, \{\{a,b\}\}, \{\emptyset, \{a\}\}, \{\emptyset, \{b\}\}, \{\emptyset, \{a,b\}\}, \{\{a\}, \{b\}\}, \{\{a\}, \{a,b\}\}, \{\{a,b\}, \{b\}\}, \{\emptyset, \{a\}, \{b\}\}, \{\emptyset, \{a\}, \{a,b\}\}, \{\emptyset, \{a,b\}, \{b\}\}, \{\emptyset, \{a\}, \{b\}, \{a,b\}\}$

    Is this correct? Or do I miss something? (Wondering)
  4. There isn't such a set, since at least the emptyset is contained in the powerset. Is this correct? (Wondering)
 

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,687
So we have the following:

1. $2^M=\{\emptyset, \{7\}, \{4\}, \{0\}, \{3\}, \{7,4\}, \{7,0\}, \{7,3\}, \{4,0\}, \{4,3\}, \{0,3\},\{7,4,0\}, \{7,4,3\}, \{7,0,3\}, \{4,0,3\}, \{7,4,0,3\}$

2. $2^{A\times B}=\text{ set of every subset of } A\times B=\{(a,b)\mid a\in A, b\in B\}$.

So it is of the form $2^{A\times B}=\{\emptyset, \{(a_i, b_i)\}, \{(a_i, b_i), (a_j, b_j)\}, \cdots \}$, or not?

This is not of the form $\{A'\times B'\mid A'\subseteq A, B'\subseteq B\}$, is it? (Wondering)

3. We have that $M=\{\emptyset, \{a\}, \{b\}, \{a,b\}\}$.

The $2^M=\{\emptyset, \{\emptyset\}, \{\{a\}\},\{\{b\}\}, \{\{a,b\}\}, \{\emptyset, \{a\}\}, \{\emptyset, \{b\}\}, \{\emptyset, \{a,b\}\}, \{\{a\}, \{b\}\}, \{\{a\}, \{a,b\}\}, \{\{a,b\}, \{b\}\}, \{\emptyset, \{a\}, \{b\}\}, \{\emptyset, \{a\}, \{a,b\}\}, \{\emptyset, \{a,b\}, \{b\}\}, \{\emptyset, \{a\}, \{b\}, \{a,b\}\}$

Is this correct? Or do I miss something?

4. There isn't such a set, since at least the emptyset is contained in the powerset. Is this correct?
All correct.
Did you miss something? Only the closing brace. (Tauri)
 

mathmari

Well-known member
MHB Site Helper
Apr 14, 2013
4,008

topsquark

Well-known member
MHB Math Helper
Aug 30, 2012
1,120
A question of my own.

Say we have \(\displaystyle A = \{1, 2, 4 \}\) and \(\displaystyle B = \{ 1, 2, 3 \}\)

Why wouldn't \(\displaystyle 2^{A \times B}\) have sets with the form \(\displaystyle \{ a_i, b_j \}\) rather than just \(\displaystyle \{ a_i, b_i \}\)?

And, since we are talking about a Cartesian product wouldn't \(\displaystyle 2^{ \{ 1, 2 \} }\) be different from \(\displaystyle 2^{ \{ 2, 1 \} }\)?

Thanks!

-Dan