Power Sets

Set Theory · Axiom Academy

The collection of all subsets of a set Given any set, we can form a new set by collecting all of its subsets together. This new set is called the power set . The power set of a set , denoted , is the set of all subsets of . The power set always includes: The empty set (the smallest subset) The set itself (the largest subset) We have 4 subsets : the empty set, two single-element subsets, and the full set. A remarkable fact: the size of a power set follows a simple pattern. If a set has elements, its power set has exactly elements. For each element in the original set, you have exactly two choices when forming a subset: With elements, that's independent yes/no decisions, giving possible combinations. There's a beautiful connection between subsets and binary numbers. Each subset corresponds to a unique binary string! For , each subset maps to a 3-bit binary number: Each bit position represents whether an element is included (1) or excluded (0). Since there are possible n-bit binary numbers (from 0 to ), there are exactly subsets! Click the bits below to toggle elements and see the corresponding subset: Complete Example: P( 1, 2, 3 ) Let's list all 8 subsets of systematically: We can organize these subsets by their size: The number of k-element subsets of an n-element set is given by the binomial coefficient . Notice that 1 + 3 + 3 + 1 = 8 = 2³! (empty set is always in the power set) (the set itself is always in the power set) (power set always has 2ⁿ elements)

This is the written version of the interactive lesson above. See the full Set Theory course.