Set Relations and Cardinality

Real Analysis · Axiom Academy

LESSON Set Relations & Cardinality Subsets, the power set, and the one clean reason a set with n elements has exactly 2^ n subsets. Start with the most basic relationship between two sets. We say A is a subset of B — written — when every element of A is also an element of B . Nothing in A is allowed to escape B . Read it literally: for every x , if x is in A then x is in B . Now collect all the subsets of a set into a single new set. The power set of S , written , is the set whose elements are exactly the subsets of S — including and S itself. Take . Watch its subsets organize themselves into a lattice by size — the empty set at the bottom, S at the top, each step up adding one element: 3. The Key Insight: A Binary Choice per Element Here is the idea that makes the count obvious. To build a subset of , walk down the list and make one independent yes/no decision per element : is it in the subset, or out ? Record each decision as a bit — 1 for in, 0 for out. The counter below runs through every 3 -bit string . For each one, the subset it names appears on the right. There are no leftovers and no repeats: 4. Cardinality and Exponential Growth The cardinality |S| is the number of elements in S . The bijection from the last step locks in the headline result, for every finite set:

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