Can you help me with this;
Find the value of $|A|$ if: $A = \{\varnothing\}\cup\varnothing$.
I am having trouble understanding null sets. Is a empty set not a subset of itself? I would be grateful some understanding of this.
Asked
Active
Viewed 118 times
1
-
For every set $B$, $B\cup\emptyset = B$. Therefore, $A={\emptyset}\cup\emptyset={\emptyset}$. Finally, $|A|=1$ since $A$ contains one element $\emptyset$ (probably by definition of $1:={\emptyset}$ depending on who you are reading). – orole Feb 05 '18 at 18:57
-
thank you this helps. so does that mean |P(A) = 1 ? – orlagh Feb 05 '18 at 19:02
-
Go here $\longrightarrow$ https://math.stackexchange.com/questions/1255726/what-is-an-empty-set?rq=1 – Mr Pie Feb 05 '18 at 19:02
-
Beware! You are probably studying the ZFC axioms, the distinction between an empty set as an object and the emptiness will be very important, and you will then be able to construct all natural numbers (and the others as well) – Lucio Tanzini Feb 05 '18 at 19:02
-
In general $|P(A)|=2^{|A|}$ ($=2^1=2$ in your case). You can write explicitly $P(A)={\emptyset, {\emptyset}}$, a one-element set has as subsets, the empty set $\emptyset$ and the whole set $A={\emptyset}$. – orole Feb 05 '18 at 19:04
-
i am unsure of the ZFC axioms, i read the post thank you. your explanations makes sense to the following - |P(A)|<|A x A| – orlagh Feb 05 '18 at 19:12
2 Answers
3
The union of $\emptyset$ with any other set is that other set. Therefore, $\{\emptyset\}\cup\emptyset=\{\emptyset\}$. So$$|A|=\bigl|\{\emptyset\}\bigr|=1.$$
José Carlos Santos
- 440,053
0
We know that set is a well-defined collection of distinct objects (source: https://en.wikipedia.org/wiki/Set_(mathematics))
Here, $\{\emptyset\}$ is a set with an object $\emptyset$ while the empty set $\emptyset = \{\}$. So the union is $\{\emptyset\} \cup \{\} = \{\emptyset\}$ and size is $1$.
In order to prevent confusion, we can simply enumerate (or rename) the object of a set, like we can say "define an object called '$a$', which is the name of the object $\emptyset$ (not the set $\emptyset$)". In this case, our set becomes $\{\emptyset\} = \{a\}$. Rest of this, I'm sure that you can do it much more easily.
ArsenBerk
- 13,388
-
if {Ø|} = 1 Then P(A)|= 2|A| =2|A| =2¹ =2 can you help me understand what the value of P(A)|< |A x A| – orlagh Feb 05 '18 at 19:43
-
Did you mean "if $|{\emptyset}| = 1$ then $|P(A)| = 2^{|A|} = 2^1 = 2$"? Because $|P(A)| = 2^{|A|}$ not $2|A|$. – ArsenBerk Feb 05 '18 at 19:45
-
If you mean that, that's correct. And actually we have $P(A) = \big{{},{\emptyset}\big}$. $P(A)$ is just a set of sets, don't get confused by that. From the definition, we can also consider sets as objects like naming the sets ${} = B$ and ${\emptyset} = C$. Then $P(A) = {B,C}$. – ArsenBerk Feb 05 '18 at 19:47
-
if |A| = {Ø} = 1 does that mean the |P(A)| = 2 and if so what would be the value of |P(A)| < |A x A| – orlagh Feb 05 '18 at 19:53
-
I don't get why people go ZFC axioms for a question as this one, it is a more advanced topic that you can learn after learning the more basic concepts in set theory. I can tell you that $A\times A = \big{{\emptyset, \emptyset}\big}$, whose size is $1$ so the argument $|P(A)| < |A \times A|$ implies $2 < 1$, which is false. – ArsenBerk Feb 05 '18 at 19:58
-
i am studying computer science in uni and these are combined study material is both not related? – orlagh Feb 05 '18 at 20:00
-
Well, for a question as this one, ZFC-axioms are unrelated (even if it is related, we don't need to use them). In general, as far as I know, there are some universities that teach ZFC-axioms in their discrete courses for computer science students. But it's more likely to be in abstract math courses so if you just began the set theory, it is early to learn ZFC-axioms in my opinion. I suggest you to learn it after you have control over more basic concepts in set theory. – ArsenBerk Feb 05 '18 at 20:04
-
i understand. i will get a better understanding your explanation helped alot thank you. – orlagh Feb 05 '18 at 20:09