Power Sets and Products

Discrete Math · Axiom Academy

LESSON Power Sets and Cartesian Products Understanding how to build new sets from existing ones: the power set contains all subsets, and the Cartesian product creates ordered pairs. 1. Power Sets: All Possible Subsets Let's build the power set step by step. Starting with a simple set A = 1, 2 , we'll visualize how each element can either be included or excluded, creating all possible subsets. The size of the power set follows a simple exponential pattern: if |A| = n, then |𝒫(A)| = 2 n . This is because for each of the n elements, we make a binary choice: include it or don't. 3. Cartesian Products: Ordered Pairs Unlike power sets which create subsets, the Cartesian product creates ordered pairs. Let's visualize A × B where A = 1, 2 and B = x, y . The size of a Cartesian product is straightforward: |A × B| = |A| × |B|. This is the multiplication principle—for each element in A, we pair it with every element in B.

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