Proving some relations between given sets

Let

$A=\{1,2\}, \quad B=\{\{1\},\{2\}\}, \quad C=\{\{1\},\{1,2\}\}, \quad D=\{\{1\},\{2\},\{1,2\}\}.$

Prove or disprove the following.

$A = B$ .
$A \subseteq B$ .
$A \subset C$ .
$A \in C$ .
$A \subset D$ .
$B \subset C$ .
$B \subset D$ .
$B \in D$ .
$A \in D$ .

False.
These sets are not equal since they do not contain the same elements (since $1 \neq \{ 1 \}$ and $2 \neq \{ 2 \}$ ).
False.
Since $A$ contains an element (e.g. 1) that is not contained in $B$ , we have $A \nsubseteq B$ . (Again, this is asking us to observe that $1 \neq \{ 1 \}$ .)
False.
Again, $1 \in A$ but $1 \notin C$ ; hence, $A \nsubseteq C$ .
True.
Proof. By definition of $C$ we have $\{ 1,2 \} \in C$ . Since $A = \{1,2\}$ , we have $A \in C$ .∎
False.
Since $1 \in A$ , but $1 \notin D$ , we have $A \not\subset D$ .
False.
The element $\{ 2 \}$ is in $B$ , but is not in $C$ . Hence, $B \not\subset C$ .
True.
Proof. To show $B \subset D$ , we must show that every element in $B$ is also in $D$ , and that $B \neq D$ (so that $B$ is a proper subset). First, the only two elements of $B$ are the sets $\{ 1 \}$ and $\{ 2 \}$ . Since each of these sets is contained in $D$ by definition of $D$ , we have $B \subseteq D$ . Finally, since the set $\{ 1,2 \} \in D$ , but $\{ 1,2 \} \notin B$ , we see that the inclusion is proper. Hence, $B \subset D$ . ∎
False.
Again, the problem is emphasizing the difference between an element and the set that contains that element. In this case $B = \{ \{1\},\{2\}\}$ and $\{\{1\}, \{2\}\}$ is not an element of $D$ (even though $\{1\}$ and $\{2\}$ are elements of $D$ , the set containing the sets $\{1\}$ and $\{2\}$ is not).
True.
Proof. By the definition of $D$ we have $\{ 1 ,2 \} \in D$ . Since $A = \{ 1,2 \}$ , we have $A \in D$ . ∎

8 comments

February 19, 2017 at 12:16 pm

Tesuji says:

For part e) {1,2} is an element of {{1},{2},{1,2}}. So it should be true.
And thanks too for your work.

February 19, 2017 at 12:11 pm

Tesuji says:

For\: part\: \left ( d \right ), \: \left \{ 1,2\right\left\right \} is\: an\: element\: of\: \left \{\left \{1\left\right \},\left \{2\left\right \},\left \{1,2\left\right\}\left. \right \}. So\: it\: should\: be\: true.

- February 19, 2017 at 12:13 pm
  
  Tesuji says:
  
  For part d) {1,2} is an element of {{1},{2},{1,2}}. So it should be true
  
  - February 19, 2017 at 12:15 pm
    
    Tesuji says:
    
    My apologies and please delete above comments.
January 7, 2016 at 9:53 pm

Anonymous says:

For part g, I think you accidentally wrote set C instead of set D starting halfway through the answer. Also, thanks for your work. Really well done.

- January 8, 2016 at 9:21 am
  
  RoRi says:
  
  Fixed now. Thanks for letting me know!
  
  - May 14, 2016 at 9:02 am
    
    Guilherme Coelho says:
    
    The same aplies to part h.
  - May 16, 2016 at 7:49 am
    
    RoRi says:
    
    Argh. Thanks! One day all of the typos and errors will get found and corrected…

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Proving some relations between given sets

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