Let A, B, and C be three sets. Then, A-B-C is not equal to

Looking at the expression $A - B - C$, this is interpreted as $(A - B) - C$.

$(A - B) - C$

$= (A \cap B^c) - C$

$= (A \cap B^c) \cap C^c$

$= A \cap B^c \cap C^c$

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Checking Option 1: $A \cap C^c \cap B^c$

This can be rewritten as:

$A \cap C^c \cap B^c = A \cap B^c \cap C^c$

This equals $A - B - C$.

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Checking Option 2: $(A - C) - B$

$(A - C) - B$

$= (A \cap C^c) - B$

$= (A \cap C^c) \cap B^c$

$= A \cap B^c \cap C^c$

This equals $A - B - C$.

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Checking Option 3: $A - B \cap C$

Following standard order of operations (intersection before difference):

$A - B \cap C = A - (B \cap C)$

$= A \cap (B \cap C)^c$

$= A \cap (B^c \cup C^c)$

This represents elements in $A$ that are either not in $B$ or not in $C$ (or both).

This is different from $A \cap B^c \cap C^c$, which requires elements to be not in $B$ AND not in $C$.

For example, an element in $A \cap B^c \cap C$ (in $A$, not in $B$, but in $C$) would be in $A \cap (B^c \cup C^c)$ but not in $A \cap B^c \cap C^c$.

This does NOT equal $A - B - C$.