Proving A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

Discrete Math · Axiom Academy

EXAMPLE Proving the Distributive Law for Sets Master the element argument technique for proving set equality through both subset inclusions. Left ⊆ Right: Every element in A ∩ (B ∪ C) is also in (A ∩ B) ∪ (A ∩ C) Right ⊆ Left: Every element in (A ∩ B) ∪ (A ∩ C) is also in A ∩ (B ∪ C) Excellent work! You've completed a rigorous proof of the distributive law for sets. Here's what we learned: Set Equality Requires Two Inclusions: To prove X = Y, we must show both X ⊆ Y and Y ⊆ X. Element Argument Method: Start with "Let x ∈ [left side]" and show x must be in [right side], then repeat in reverse. Apply Definitions Carefully: Intersection (∩) means "and" (both conditions), union (∪) means "or" (at least one condition). Case Analysis: When dealing with unions, consider each case separately to handle all possibilities. Logical Structure Matters: Each step must follow logically from the previous ones using definitions and established facts. Both Directions Are Essential: Never assume one direction proves the other—both must be shown independently. This proof technique is fundamental in discrete mathematics and will be used throughout set theory, logic, and many other areas. Practice identifying when to use case analysis and how to structure your proofs clearly!

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