Complement and Difference

Set Theory · Axiom Academy

LESSON Complement and Difference So far we've learned to combine sets (union, intersection). Now we'll learn to remove elements. There are two key operations: Complement: Everything outside a set (relative to a universal set) Difference: Remove one set's elements from another To talk about complements, we need a universal set that contains all objects under consideration. The complement of A consists of everything in U that isn't in A. The complement of a set A, written , is the set of all elements in the universal set U that are not in A. Aᶜ = Everything shaded (outside A, inside U) The complement contains all integers from 1 to 10 that aren't even. You may see the complement written as: The difference of sets A and B, written , is the set of elements that are in A but not in B. A \ B = Only the shaded part (in A but not B) Note: A \ B ≠ B \ A in general! Set difference is not commutative. Set difference is sometimes written as (using minus sign). So . Complement and difference are closely related: The complement is just the difference from the universal set! Taking "A but not B" is the same as "A and (not B)" Two of the most important laws in set theory connect complement with union and intersection: "Not (A or B)" = "(not A) and (not B)" "Not (A and B)" = "(not A) or (not B)" Visualizing (A ∪ B)ᶜ = Aᶜ ∩ Bᶜ The shaded region is outside both A and B Convert between unions and intersections Simplify complex set expressions

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