# Probability Question

## 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.

HallsofIvy
Homework Helper
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

rollingstein
Gold Member
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

HallsofIvy
Homework Helper
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?

haruspex
Homework Helper
Gold Member
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 "..." ?

HallsofIvy
Homework Helper
$B^{c} \subseteq A^{c}$

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!