To prove \( a \cap b = (b - a) \Delta b \), let's carefully analyze both sides of the equation and the operations involved.

1. **Given Sets and Operations:**
   - \( a \cap b \): This represents the intersection of sets \(a\) and \(b\), which consists of elements that are in both \(a\) and \(b\).
   - \( b - a \): This is the set difference, which consists of elements in \(b\) but not in \(a\).
   - \( \Delta \): This is the symmetric difference operation. For sets \(x\) and \(y\), the symmetric difference \( x \Delta y \) is defined as \( (x - y) \cup (y - x) \), meaning elements in either \(x\) or \(y\), but not in both.

2. **Expanding the Right Side:**
   - Let's rewrite the right side using the definition of the symmetric difference:
   \[
   (b - a) \Delta b = (b - a - b) \cup (b - b + a)

Explore the step-by-step proof of the set equation a ∩ b = (b - a) ∆ b. This problem covers essential set theory concepts, including intersection, set difference, and symmetric difference operations. Suitable for students with an understanding of advanced set theory, typically in undergraduate mathematics or computer science.

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