# Thread: Null value in the context of sets

1. ## Null value in the context of sets

Hi,

I hope someone can help. I'm trying to figure out whether the following statements are true or false:

1) ∅ ⊂
S for all sets S
I would say that this is false since we don't know what is contained within the set of S. But perhaps since we don't know, we assume that it contains nothing and therefore it is true?

2) ∅ ∩ S = ∅ for all sets S
This statement would suggest that S contains elements, and so therefore the answer is null since
∅ never overlaps with the set S. Again, I don't really know what the value of S represents, so I'm left here to assume that elements exist in the set - so I'm not going to say anything about the statement until I have a better sense of what S is.

For both questions I would really like to understand what is contained within the set of S. If I knew that, I probably would be able to determine its correct boolean value.

Let me know,
- Olivia

2. ## Re: Null value in the context of sets

Originally Posted by otownsend
1) ∅ ⊂ S for all sets S

2) ∅ ∩ S = ∅ for all sets S
These are just an exercise in pure logic.
Do you see that If $1+1=4$ then $4=3$ is necessarily TRUE ?
If you do not, then these proofs will be meaningless.

The set that has on elements is denoted by $\emptyset$.

The statement that $x\in\emptyset$ is necessarily false. WHY?

The statement that $A\subseteq B$ means that if $x\in A$ then $x\in B$.

Now the proof.

If $x\in\emptyset$ then $x\in A$ must be true. WHY?
So that means $\emptyset\subseteq A$ by definition.

3. ## Re: Null value in the context of sets

Originally Posted by otownsend
[FONT="]Hi,

I hope someone can help. I'm trying to figure out whether the following statements are true or false:

1) ∅ ⊂
[/FONT]
[FONT="]S[/FONT][FONT="] for all sets [/FONT][FONT="]S
I would say that this is false since we don't know what is contained within the set of S. But perhaps since we don't know, we assume that it contains nothing and therefore it is true?

2) ∅ ∩ S = ∅ for all sets S
This statement would suggest that S contains elements, and so therefore the answer is null since
[/FONT]
∅ never overlaps with the set S. Again, I don't really know what the value of S represents, so I'm left here to assume that elements exist in the set - so I'm not going to say anything about the statement until I have a better sense of what S is.

For both questions I would really like to understand what is contained within the set of S. If I knew that, I probably would be able to determine its correct boolean value.

Let me know,
- Olivia
1) $\text{for any set }A \text{, }A \subset S \text{ if }\forall x \in A, x \in S$

$\text{now let }A=\phi \\ \\ \phi \text{ has no elements, so }\forall x \in \phi, x \in S,\text{ and thus } \phi \subset S$

2) $\text{given any two sets, }A,~B,~A\cap B \subset A,~A\cap B \subset B$

$\text{so for any set S, }\phi \cap S \subset \phi$

$\text{there is only one set that is a subset of }\phi \text{ and that is }\phi \text{ itself.}$

$\text{thus }\phi \cap S = \phi$

4. ## Re: Null value in the context of sets

Originally Posted by otownsend
[FONT="]Hi,

I hope someone can help. I'm trying to figure out whether the following statements are true or false:

1) ∅ ⊂
[/FONT]
[FONT="]S[/FONT][FONT="] for all sets [/FONT][FONT="]S.
I would say that this is false since we don't know what is contained within the set of S. But perhaps since we don't know, we assume that it contains nothing and therefore it is true?[/I][

No. saying that a set, S, has the empty set as a subset does not say that S itself is empty. For example, the set {1, 2}, since it has 2 members, has $\displaystyle 2^2= 4$ subsets. They are: {1, 2}, {1}, {2}, and {}.
(It can be proved that any finite set with n elements has $\displaystyle 2^n$ subsets, including the empty set and the entire set itself.

2) ∅ ∩ S = ∅ for all sets S
This statement would suggest that S contains elements, and so therefore the answer is null since [/I][/FONT]
∅ never overlaps with the set S. Again, I don't really know what the value of S represents, so I'm left here to assume that elements exist in the set - so I'm not going to say anything about the statement until I have a better sense of what S is.

No, that is not what this means. This is saying that every set has the empty set as a subset- it doesn't matter what set S is. And it does not "suggest that S contains elements". $\displaystyle \{ \}\cap \{ \}= \{ \}$ .

For both questions I would really like to understand what is contained within th[e set of S. If I knew that, I probably would be able to determine its correct boolean value.

Let me know,
- Olivia
What you need to understand are the definitions of "subset" and "intersection".

1) A set X is a "subset" of set S if and only if the statement "if $\displaystyle a\in X$ then $\displaystyle x\in S$" is true.
If X is empty then the hypothesis "if $\displaystyle a\in X$" is false so that statement is "trivially true". Another way of looking at it is that "X is not a subset of S only if X contains some member, a, that is not in S. If X is empty that is obviously impossible.

Those are true for all sets S- you do not need to know what is in S.

2) The intersection of two sets, $\displaystyle S\cap X$, consists of all members of S that are also members of X. If X itself is empty then there are no elements that are in both sets because there are not elements in X. Look at the example S= {1, 2} again. $\displaystyle \{1, 2\}\cap \{ \}$ consists of all elements that in both {1, 2} and {}. What are they? There is nothing in {} so there can be nothing in both. $\displaystyle \{1, 2\}\cap \{ \}= \{ \}$.

5. ## Re: Null value in the context of sets

That makes sense so THANK YOU! Again, I appreciate your help.