Venn Diagrams and Partitions Chapter 1-2 Venn Diagrams and Partitions

Venn Diagrams In a Venn Diagram, a universal set U and its subsets are pictured by using geometric shapes. The universal set is usually represented by a rectangle and the subsets usually circles inside the rectangle.

Venn Diagrams A B A B A B A B A’

Venn Diagrams A B A B A B A B A’

Venn Diagrams A B A B 𝐴∪𝐵 𝐴 ′ ∩𝐵′ A B A B 𝐴∩𝐵 𝐴′∪𝐵′

Venn Diagrams A B A B 𝐴∪𝐵 𝐴 ′ ∩𝐵′ A B A B 𝐴∩𝐵 𝐴′∪𝐵′

deMorgan’s Laws For any subsets A and B of a universal set U 𝐴∪𝐵 ′ =𝐴′∩𝐵′ “The complement of a union is the intersection of the complements” 𝐴∩𝐵 ′ =𝐴′∪𝐵′ “The complement of an intersection is the union of the complements” Other useful relations: 𝐴∩ 𝐵∪𝐶 =(𝐴∩𝐵)∪(𝐴∩𝐶) 𝐴∪ 𝐵∩𝐶 =(𝐴∪𝐵)∩(𝐴∪𝐶) The parentheses are essential

Why Parentheses Matter 𝐴∩ 𝐵∪𝐶 (𝐴∩𝐵)∪𝐶

Example Let 𝑋= 2,4,6,8,10,11,12,13,14,16,18 Find subsets 𝑋 1 , 𝑋 2 , 𝑋 3 such that 𝑋 1 contains all even numbers in X less than 10, 𝑋 2 contains all odd numbers in X, and 𝑋 3 contains all even numbers in X at least as large as 10. 𝑋 1 = 𝑋 2 = 𝑋 3 = Note that 𝑋= *intersections: Pairwise Disjoint: every pair of sets in the collection is disjoint A Partition of a set X is a collection of nonempty subsets of X which are pairwise disjoint and whose union is the entire set X

Counting Let A be a set with a finite number of elements. The number of elements in A is denoted 𝑛(𝐴). For instance, if X, 𝑋 1 𝑋 2 and 𝑋 3 are subsets, then,𝑛 𝑋 =11, 𝑛 𝑋 1 =4, 𝑛 𝑋 2 =2, 𝑛 𝑋 3 =5 𝑛 𝑋 = 𝑛 𝑋 1 +𝑛 𝑋 2 +𝑛 𝑋 3

Partition Principal If a set X is partitioned into subsets 𝑋 1, 𝑋 2, . . . . , 𝑋 𝑘, then 𝑛 𝑋 =𝑛 𝑋 1 +𝑛 𝑋 2 +. . . . + 𝑛 𝑋 𝑘 If each subset 𝑋 1, 𝑋 2, . . . . , 𝑋 𝑘, has the same number of elements, then we can simplify it to 𝑛 𝑋 =𝑘𝑛 𝑋 1

Example A student is to plan a schedule which consists of one science course and one humanities course. The student can choose a science course in Astronomy (A), Biology (B), or Chemistry (C) and a humanities course in History (H), Philosophy (P), Religion (R), or theatre (T). Determine the number of possible schedules. 𝑛 𝑆 =𝑛 𝐴,𝐵,𝐶 ∙ 𝑛 𝐻,𝑃,𝑅,𝑇 =3∙4=12 If A and B are sets, then 𝑛 𝐴×𝐵 =𝑛(𝐴)∙𝑛(𝐵)

Example Suppose that the student in the last example also plans to take one language course, either French (F) or German (G), then the possible class schedules can be represented by ordered triples 𝑥,𝑦,𝑧 . 𝑆= 𝐴,𝐵,𝐶 × 𝐻,𝑃,𝑅,𝑇 × 𝐹,𝐺 𝑛 𝑆 =𝑛 𝐴,𝐵,𝐶 ∙ 𝑛 𝐻,𝑃,𝑅,𝑇 ∙𝑛( 𝐹,𝐺 )=3∙4∙2=24 In general: 𝑋 1, 𝑋 2, . . . . , 𝑋 𝑘, are sets 𝑛 𝑋 1 × 𝑋 2 ×∙ ∙ ∙× 𝑋 𝑘, =𝑛 𝑋 1 ∙𝑛 𝑋 2 ∙ . . . ∙𝑛( 𝑋 𝑘 )