- #1

GreenPrint

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## Homework Statement

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

## Homework Equations

De'Morgan's law

[itex](A \bigcup B)^{c} = (A^{c} \bigcap B^{c})[/itex]

## The Attempt at a Solution

A[itex]\bigcap (A \bigcup B)^{c}[/itex]

I use De'Morgan's Law

[itex]A \bigcap (A^{c} \bigcap B^{c})[/itex]

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

[itex](A \bigcap A^{c}) \bigcap (A \bigcap B^{c})[/itex]

If I'm not mistaken

[itex]A \bigcap A^{c}[/itex] = 1

so

[itex] 1 \bigcap (A \bigcap B^{c})[/itex]

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

[itex]A \bigcap B^{c}[/itex]

So I guess the answer is when

[itex] A \bigcap B^{c} = ∅ [/itex]

Does this look right?

Thanks for any help!