# Probability Q: Show B^c Subset of A^c

• GreenPrint
In summary: Definitions are what help us understand theorems. Without definitions we would just be guessing.In summary, B^{c} \subseteq A^{c}, demonstrating that B\subseteq A and that B^c\subseteq A^c.
GreenPrint

## Homework Statement

Suppose that for two events A and B. A $\subseteq$ B. Show that $B^{c} \subseteq A^{c}$.

## The Attempt at a Solution

$1- B \subseteq 1- A$

I'm not sure where to go from here. Thanks for any help you can provide.

That's really a "set" problem rather than a probability problem. I presume your "1" represents the "universal set" rather than a number?

The "standard" way to prove "$X\subset Y$" is to start "let $x\in X$" and, using the definitions and information you are given about X and Y, end with "therefore $x\in Y$". Here, "X" is the set $B^c$, the complement of B, the set of all objects that are not in B.

So, if $x\in B^c$, then x is not in B. Since A is a subset of B (every member of A is also in B), x is not in A and so $x\in A^c$.

Hi HallsofIvy,

Is universal set and null set the same?

thanks,

green

GreenPrint said:
Hi HallsofIvy,

Is universal set and null set the same?

No. Not the same.

You mean by universal set Ω?

Also how does showing that
x$\in B^{c}$ and x$\in A^{c}$

show that

$B^{c} \subseteq A^{c}$?

I'm not seeing this

Okay, it sounds like you are using $\Omega$ to mean the "universal set", the set of all things we are allowing in our sets.

It is NOT true that "$x\in B^c$ and $x\in A^c$" implies $B^c\subseteq A^c$" and I did not claim it was. What I showed was that if$x\in B^c$ implies $x\in A^c$, then $B^c\subseteq A^c$.

That follows directly from the definition of "$\subset$". If you are doing problems like that you surely should know that definition.

$B^{c} \subseteq A^{c}$

This means that B complement is a subset or equal to A complement. If x is an element of B complement and A complement than why must B complement be a subset or equal to A complement? How do I know that A complement is not a subset of B complement?

Since x is an element of A complement and B complement then...

B complement must be a subset or equal to A complement
or
A complement must be a subset or equal to B complement

how do I know which one is true?

GreenPrint said:
If x is an element of B complement and A complement than why must B complement be a subset or equal to A complement? How do I know that A complement is not a subset of B complement?

Since x is an element of A complement and B complement then...

B complement must be a subset or equal to A complement
or
A complement must be a subset or equal to B complement
None of that makes sense. If some x is an element of B complement and A complement then all that proves is that their intersection is non-empty. You need to think in terms "if x is an element of ... then x is an element of ..." You're trying to prove that Bc ⊆ Ac. Can you fill in the "..." ?

GreenPrint said:
$B^{c} \subseteq A^{c}$

This means that B complement is a subset or equal to A complement. If x is an element of B complement and A complement than why must B complement be a subset or equal to A complement?
I didn't say that! A single "x" tells us nothing. The point was that this is true for all members of B complement. That is the definition of "$X\subseteq Y$": for all $x\in X$, $x\in Y$.

How do I know that A complement is not a subset of B complement?

Since x is an element of A complement and B complement then...

B complement must be a subset or equal to A complement
or
A complement must be a subset or equal to B complement

how do I know which one is true?
By knowing what "subset" and "member" mean! Go back and look at my argument, showing that any element of B complement must be a member of A complement.

And that is the definition of "B complement is a subset of A complement".

If your are asking, as you appear to be, "How can I prove this without knowing any of the definitions", you can't!

## 1. What is the definition of a complement in probability?

A complement in probability refers to the set of all outcomes that are not included in a given event. It is denoted by adding a superscript "c" to the event. For example, the complement of event A is written as Ac.

## 2. How is a complement related to the concept of probability?

A complement is closely related to the concept of probability as it represents the probability of an event not occurring. The probability of an event and its complement always add up to 1, as they encompass all possible outcomes.

## 3. What does it mean for Bc to be a subset of Ac?

This means that all outcomes that are not in event B are also not in event A. In other words, the complement of B is contained within the complement of A.

## 4. How does this relate to the probability of Bc and Ac occurring?

If Bc is a subset of Ac, this means that the probability of Bc occurring is less than or equal to the probability of Ac occurring. This is because Ac includes more outcomes than Bc, making it a less specific event.

## 5. Can you provide an example to illustrate this concept?

Sure, let's say we have a bag with 10 marbles, 5 red and 5 blue. If we define event A as selecting a red marble and event B as selecting a blue marble, then Ac would be selecting a blue marble and Bc would be selecting a red marble. Since Bc is a subset of Ac, the probability of selecting a blue marble (Bc) is less than or equal to the probability of selecting a red marble (Ac).

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