# Solving Probability Question: Empty Intersection Condition

• GreenPrint
In summary, the empty intersection condition is when two events have no outcomes in common, affecting probability calculations by resulting in a lower probability value. It cannot occur with independent events and can be used in real-world scenarios to identify and eliminate impossible outcomes. It is also related to the concept of complements in probability.
GreenPrint

## Homework Statement

Let A and B be two events in a sample space. Under what condition(s) is A$\bigcap (A \bigcup B)^{c}$ empty?

## Homework Equations

De'Morgan's law

$(A \bigcup B)^{c} = (A^{c} \bigcap B^{c})$

## The Attempt at a Solution

A$\bigcap (A \bigcup B)^{c}$

I use De'Morgan's Law

$A \bigcap (A^{c} \bigcap B^{c})$

I don't know if I can do this or not but I think it's what I'm supposed to do.

$(A \bigcap A^{c}) \bigcap (A \bigcap B^{c})$

If I'm not mistaken

$A \bigcap A^{c}$ = 1

so

$1 \bigcap (A \bigcap B^{c})$

If I'm not mistaken this can be simplified some more to

$A \bigcap B^{c}$

So I guess the answer is when

$A \bigcap B^{c} = ∅$

Does this look right?

Thanks for any help!

GreenPrint said:

## Homework Statement

Let A and B be two events in a sample space. Under what condition(s) is A$\bigcap (A \bigcup B)^{c}$ empty?

## Homework Equations

De'Morgan's law

$(A \bigcup B)^{c} = (A^{c} \bigcap B^{c})$

## The Attempt at a Solution

A$\bigcap (A \bigcup B)^{c}$

I use De'Morgan's Law

$A \bigcap (A^{c} \bigcap B^{c})$

I don't know if I can do this or not but I think it's what I'm supposed to do.

$(A \bigcap A^{c}) \bigcap (A \bigcap B^{c})$

If I'm not mistaken

$A \bigcap A^{c}$ = 1

so

$1 \bigcap (A \bigcap B^{c})$

If I'm not mistaken this can be simplified some more to

$A \bigcap B^{c}$

So I guess the answer is when

$A \bigcap B^{c} = ∅$

Does this look right?

Thanks for any help!

1? What kind of a set is 1? Isn't $A \bigcap A^{c} = \phi$, the empty set?

GreenPrint said:
If I'm not mistaken $A \bigcap A^{c}$ = 1
But unfortunately you are mistaken. Is it what you intended to write... that (in the language of set theory) the intersection of a set with its complement is the universal set?

What do you mean by universal set? Now that I think of it I think perhaps empty set is the answer to that. Isn't it empty under all conditions since

∅ = A$\bigcap A^{c}$

GreenPrint said:
What do you mean by universal set? Now that I think of it I think perhaps empty set is the answer to that.

Yes, it is. It's empty regardless of what B is.

GreenPrint said:
What do you mean by universal set? Now that I think of it I think perhaps empty set is the answer to that. Isn't it empty under all conditions since

∅ = A$\bigcap A^{c}$
Given a set, A, Ac is the set of all objects that are NOT in A. But what is meant by "all objects"? In order that we not have to include nonsensical things like "Jupiter's fourth moon" or "the fairies that live down by the creek", we have to have a specific "domain of discourse"- all those things that we are talking about. The set containing all things that in set A or B or, in fact, all the things we allow to be in the sets we are talking about is the "universal set". You cannot talk about the "complement of a set" without having a "universal set" so I suspect you just know it by a different name.

Last edited by a moderator:
Isn't that always empty?

I used a Venn Diagram

Isn't what always empty? If you are referring to the original post, $A\cap (A \cup B)^c$, yes, that is empty no matter what A and B are. In order to be in $A\cap (A\cup B)^c$, x must be in both A and $(A\cup B)^c$. In order to be in $(A\cup B)^c$, x must not be in $A\cup B$. But $A\cup B$ includes all members of A so that x cannot be in A. There is NO x in $A\cap (A\cup B)^c$.

Last edited by a moderator:
HallsofIvy said:
Isn't what always empty? If you are referring to the original post

Yes. Thanks.

## 1. What is the empty intersection condition?

The empty intersection condition refers to a scenario in probability where two events have no outcomes in common. This can occur when the events are mutually exclusive or when they have no overlap in their sample spaces.

## 2. How does the empty intersection condition affect probability calculations?

The empty intersection condition affects probability calculations by making the intersection of the two events equal to 0, which in turn affects the overall probability calculation. This often results in a lower probability value.

## 3. Can the empty intersection condition occur with independent events?

No, the empty intersection condition cannot occur with independent events. Independent events have no influence on each other, so there will always be at least one outcome in common between them.

## 4. How can the empty intersection condition be used in real-world scenarios?

The empty intersection condition can be used in real-world scenarios to identify and eliminate impossible outcomes from a set of events. This can help in decision-making processes where certain outcomes are not feasible or realistic.

## 5. Are there any other mathematical concepts related to the empty intersection condition?

Yes, the empty intersection condition is closely related to the concept of complements in probability. The complement of an event is the set of all outcomes that are not part of the event. In the case of the empty intersection condition, the complement of one event is the entire sample space of the other event.

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