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

[GreenPrint]

Homework Statement

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

Homework Equations

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]

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 $x\in A^c$.

 

[GreenPrint]

Hi HallsofIvy,

Is universal set and null set the same?

thanks,

green

 

[rollingstein]

Hi HallsofIvy,

Is universal set and null set the same?

No. Not the same.

 

[GreenPrint]

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]

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.

 

[GreenPrint]

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

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]

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 B^c ⊆ A^c. Can you fill in the "..." ?

 

[HallsofIvy]

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

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!