Introduction to Venn diagrams
Partitions
Counting principles
§1.2 Venn Diagrams and Partitions
Tom Lewis
Fall Semester
2014
Introduction to Venn diagrams
Partitions
Outline
Introduction to Venn diagrams
Partitions
Counting principles
Counting principles
Introduction to Venn diagrams
Partitions
Counting principles
Example
• Draw the Venn diagrams for A ∩ B, A ∪ B and A 0 .
• Verify DeMorgan’s law
(A ∪ B) 0 = A 0 ∩ B 0
and (A ∩ B) 0 = A 0 ∪ B 0 .
using Venn diagrams.
• Verify the distributive law
A ∩ (B ∪ C ) = (A ∩ B) ∪ (A ∩ C ).
Introduction to Venn diagrams
Partitions
Figure : Two ways to organize your coins
Counting principles
Introduction to Venn diagrams
Partitions
Counting principles
Definition
The collection of sets A1 , A2 , . . . , An is said to be pairwise disjoint
provided that every pair of sets is disjoint.
Example
Let A = {1, 2, 3}, B = {4, 5, 6}, C = {7, 8, 9, 10}. These sets are
pairwise disjoint.
Introduction to Venn diagrams
Partitions
Counting principles
Definition
A partition of the set X is a collection A1 , A2 , . . . , An of nonempty
subsets of X which are pairwise disjoint and whose union is X .
Problem
Partition the set X = {1, 2, 3, 4, 5, 6, 7, 8} into three sets.
Introduction to Venn diagrams
Partitions
Counting principles
Definition
Given a set A with a finite number of elements, let
n(A) = the number of elements in A.
Example
If A = {a, b, c, d , e}, then n(A) = 5.
Introduction to Venn diagrams
Partitions
Counting principles
Theorem (The Partition Principle)
Let A1 , . . . , Ak denote a partition of X , then
n(X ) = n(A1 ) + · · · + n(Ak ).
In other words the number of elements in the set X can be found
by adding the numbers of elements in each of the partitioning sets.
Introduction to Venn diagrams
Partitions
Counting principles
Problem
Consider the following data. Within a certain dormitory, there are
30 students taking Bio 11, 35 students taking Math 16, and 15
students taking both Bio 11 and Math 16. How many of these
students are taking either Bio 11 or Math 16.
Introduction to Venn diagrams
Partitions
Counting principles
Theorem (Counting Principle for Product Sets)
Let A and B be finite sets. Then
n(A × B) = n(A)n(B).
Theorem (A General Counting Principle for Product Sets)
In general, let A1 , A2 , . . . , Ak be a collection of finite sets. Then
n(A1 × A2 × · · · × Ak ) = n(A1 )n(A2 ) · · · n(Ak ).
Introduction to Venn diagrams
Partitions
Counting principles
Problem
Toss a red die and a blue die and record the outcome in the form
(r , b), where r is the ouctome of the red die and b is the outcome
of the blue die. How many outcomes are there for this experiment?