Illustrate [A-(B U C)] intersection [B - (A U C)] intersection [C-(A U B)] using a Venn Diagram. Explain the result by simplifying the expression analytically.

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- Jun 18th 2008, 01:22 AMazuresonataSets and Venn Diagram problem
Illustrate [A-(B U C)] intersection [B - (A U C)] intersection [C-(A U B)] using a Venn Diagram. Explain the result by simplifying the expression analytically.

Thanks - Jun 18th 2008, 06:14 AMkalagota
if $\displaystyle x \in [A - (B \cup C)]$, then $\displaystyle x \in A$ and $\displaystyle x \in (B \cup C)^c$..

thus, $\displaystyle x \in A$ and $\displaystyle x \in (B^c \cap C^c)$ and hence $\displaystyle x \in A$ and $\displaystyle x \in B^c$ and $\displaystyle x \in C^c$

so if $\displaystyle x \in [A - (B \cup C)] \cap [B - (A \cup C)] \cap [C - (A \cup B)]$ then $\displaystyle x \in [A - (B \cup C)]$ and $\displaystyle x \in [B - (A \cup C)]$ and $\displaystyle x \in [C - (A \cup B)]$

upon "distribution", you will come up with $\displaystyle x$ inside all these sets: $\displaystyle A, B, C, A^c, B^c, C^c$ (which cannot happen) which will give you a conclusion that $\displaystyle x$ must be in the empty set.. (But duh! something is inside the empty set! (Surprised)

determine which is which in the attachment..