1. ## quick question about set

Can you please check if they are correct?
a. $\emptyset \subset${0} <-- True
b. $\emptyset \subset${ $\emptyset$,{0}} <-- True
c. { $\emptyset$} $\in${ $\emptyset$} <-- True
d. { $\emptyset$ } $\in$ {{ $\emptyset$ }} <-- False
e. { $\emptyset$} $\subset$ { $\emptyset$ , { $\emptyset$}} <-- True
f. {{ $\emptyset$}} $\subset$ { $\emptyset$,{ $\emptyset$}} <--False
g. {{ $\emptyset$}} $\subset$ {{ $\emptyset$},{ $\emptyset$}} <--False

Thank you

2. Hello zpwnchen
Originally Posted by zpwnchen
Can you please check if they are correct?
a. $\emptyset \subset${0} <-- True True, the empty set is a subset of every set
b. $\emptyset \subset${ $\emptyset$,{0}} <-- True True
c. { $\emptyset$} $\in${ $\emptyset$} <-- True No, it's false. $\color{red}\{\emptyset\}=\{\emptyset\}$
d. { $\emptyset$ } $\in$ {{ $\emptyset$ }} <-- False No, it's true. It is always true that $\color{red}\{x\}\in\{\{x\}\}$ whatever $\color{red}x$ may stand for.
e. { $\emptyset$} $\subset$ { $\emptyset$ , { $\emptyset$}} <-- True True
f. {{ $\emptyset$}} $\subset$ { $\emptyset$,{ $\emptyset$}} <--False No, it's true: $\color{red}\{\{x\}\}\subset\{\emptyset,\{x\}\}$
g. {{ $\emptyset$}} $\subset$ {{ $\emptyset$},{ $\emptyset$}} <--False Correct, it is false, it should be $\color{red}\{\{\emptyset\}\}\subseteq\{\{\emptyset \},\{\emptyset\}\}$ or $\color{red}\{\{\emptyset\}\}=\{\{\emptyset\},\{\em ptyset\}\}$

Thank you

3. Originally Posted by zpwnchen
Can you please check if they are correct?
[snip]
g. {{ $\emptyset$}} $\subset$ {{ $\emptyset$},{ $\emptyset$}} <--False

Thank you

The answer depends on your interpretation of the symbol $\subset$. Some authors define this symbol to mean either proper inclusion or equality of sets. I.e., $A \subset B$ means every element of A is also an element of B. If your book or teacher defines it this way then your answer is wrong.

Other authors define $\subset$ to mean proper inclusion, i.e. $A \subset B$ means every element of A is also an element of B but $A \neq B$, and write $A \subseteq B$ or $A \subseteqq B$ if A may be a proper subset of B or equal to B. So it depends.

Hello zpwnchen

$
\color{red}\{\{\emptyset\}\}=\{\{\emptyset\},\{\em ptyset\}\}
$

$
\color{red}\{\{\emptyset\}\}\subseteq\{\{\emptyset \},\{\emptyset\}\}
$

Why they are equal? there are two elements in B, one in A. so it means every elements of A is not every elements of B...so should be $
A \neq B
$
right... ?sorry a little bit confused

5. Originally Posted by zpwnchen
$
\color{red}\{\{\emptyset\}\}=\{\{\emptyset\},\{\em ptyset\}\}
$

$
\color{red}\{\{\emptyset\}\}\subseteq\{\{\emptyset \},\{\emptyset\}\}
$

Why they are equal? there are two elements in B, one in A. so it means every elements of A is not every elements of B...so should be $
A \neq B$
By definition for sets $P=Q$ if and only if $P \subseteq Q\;\& \;Q \subseteq P$.

Therefore $\{a\}=\{a,a\}$ because $\{a\}\subseteq \{a,a\}\;\& \;\{a,a\} \subseteq \{a\}$

So $\{\{\emptyset\}\}=\{\{\emptyset\},\{\emptyset\}\}
$

BTW: The set $\{a,a\}$ has only one element.

Hello zpwnchen

Originally Posted by Plato
By definition for sets $P=Q$ if and only if $P \subseteq Q\;\& \;Q \subseteq P$.

Therefore $\{a\}=\{a,a\}$ because $\{a\}\subseteq \{a,a\}\;\& \;\{a,a\} \subseteq \{a\}$

So $\{\{\emptyset\}\}=\{\{\emptyset\},\{\emptyset\}\}
$

BTW: The set $\{a,a\}$ has only one element.
Thank you
$\emptyset \in \{x\}$
$\emptyset \subset \{x\}$
$\emptyset \subseteq \{x\}$

can you please explain me why these are true or false?

7. Originally Posted by zpwnchen
Thank you
$\emptyset \in \{x\}$ False
$\emptyset \subset \{x\}$ True
$\emptyset \subseteq \{x\}$ True
can you please explain me why these are true or false?
You, yourself, have noted that the emptyset is a subset of every set.
But in the first case you have the element symbol and usually we understand that $x \ne \emptyset$

8. Thank you!
A = {0,1,2,3,4,5,6,7,8,9}
B = {0,2,4,6,8}
can we say that A $\subset$ B and A $\subseteq$ B? But A $\ne$ B, right?

9. Originally Posted by zpwnchen
Thank you!
can we say that A $\subset$ B and A $\subseteq$ B? But A $\ne$ B, right?
Well $A \not\subset B$. A contains elements that are not in B.
But $B \subset A$, every element in B is in A.
can we say that B $\subset$ A and B $\subseteq$ A? But A $\ne$ B, right?