Maths: How Measurement Error Blurs Set Boundaries

Set theory gives us a precise language for sorting a population into categories, and Venn diagrams make those relationships visible at a glance. With a universal set U of 200 applicants and two overlapping sets A and B, the key quantities are the region sizes: only A, only B, their intersection, and the complement outside both. These connect through simple subtraction — only A is |A| minus |A ∩ B|, and the neither region follows from subtracting all inner regions from |U|. What makes this more than bookkeeping is measurement uncertainty. When every score carries an error of ±5%, a threshold like 700 is not a sharp line but a band, since 700 × 0.95 = 665 and 700 × 1.05 = 735. Applicants near that boundary can shift between regions depending on the occasion, producing false positives and false negatives. The region counts therefore look exact while resting on shaky ground, and recognising that gap between clean diagrams and messy data is the real insight.