Proof of sets

J

JellyFish

Junior Member
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Jan 12, 2009
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51
Just wondering if anyone could help me out with the follinwg problems...

1. For sets A, B and C, prove that (A?C) ? B ? (A ? B)?C . and

2. Prove that there is equality in (1) if and only if B?C = ?. For this part I know that X ?Y = ? iff X ? Y.

Thanks so much.
 
\(\displaystyle \begin{array}{rcl} {\left( {A \cup C} \right)\backslash B} & = & {\left( {A \cup C} \right) \cap B^c } \\ {} & = & {\left( {A \cap B^c } \right) \cup \left( {C \cap B^c } \right)} \\ {} & \subseteq & {\left( {A\backslash B} \right) \cup C} \\ \end{array}\)
 
Whats confusing is you wrote: For this part I know that X ?Y = ? iff X ? Y.

That is NOT true.

For the proof, you need to show => and <= directions.

For =>, assume equality holds and show that B intersect C is empty. You might want to try assuming BWOC ("by way of contradiction") that there is an element contained in the intersection and try to contradict one of your assumptions.

For <=, assume B and C contain no mutual elements. Show, in a similar manner as pka's proof, that the subset sign used on the last line is actually equality. Hint: If B and C contain no mutual elements then C is a subset of B's complement.
 
I worked out => just fine but I am still struggling with <=.

Thanks for all of your help again.
 
Again, take the second-to-last line of what pka posted.

Since B ? C = ? we know B[sup:3du9nlqh]c[/sup:3du9nlqh] ? C = C. Hence (A ? B[sup:3du9nlqh]c[/sup:3du9nlqh]) ? (C ? B[sup:3du9nlqh]c[/sup:3du9nlqh]) = (A \ B) ? C
 
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