MDM4U
Sets and Venn Diagrams
EXPECTATIONS - use Venn diagrams as a tool for organizing information in counting problems We often use sets to describe situations where order is not important. That is, T, H, L and T, L, H are different arrangements, but {T, H, L} and {T, L, H} are the same sets. Set Notation EX1. Consider the set of all prime numbers less than 20 and the set of all odd one digit numbers. these sets in set notation. Write
The notation n(X) gives the number of elements in set X. Here,
A c B is the set of elements contained in set A OR set B. In this case, A c B =
A 1 B is the set of elements contained in set A AND set B. In this case, A 1 B =
For each situation, there is a universal set, S, that contains all elements for the situation. Here, S might be the set of all natural numbers (i.e. S = ù) or perhaps, the set of all natural numbers less than or equal to twenty. A graphical representation of our sets: We usually don’t list all elements. The final Venn diagram shows only the number of elements. We would draw:
If we are interested in the elements NOT in the set X (called the complement of X), we write X’. From our example, B’ = and (A c B)’ = and n(B’) =
EX2. Consider the following sets: S is the set of the standard deck of 52 cards A is the set of diamonds B is the set of red face cards C is the set of black jacks List, find or otherwise describe each of the following sets or values. a. n(C) b. A’ c. A 1 B d. n(A c C) e. n(A c B)
Other “sets” tid-bits If sets A and B do not share any elements, we say set A and set B are disjoint sets. Also, we say A and B are mutually exclusive events. (Note: n(A 1 B) = 0) x 0 A means x is an element in the set A. A f B means A is a subset of set B. All elements of set A are also elements of set B. i or {} represents the empty set or null set. It has no elements. EX3. a. b. c. d. e. f. g. Given the Venn diagram shown, state: a set that is a subset of another set two disjoint sets n(D) n(C) n(A 1 B) n(B 1 C) n(A 1 B 1 C)
Principle of Inclusion and Exclusion EX3. The NWSS girls volleyball team has 13 members, and the girls basketball team has 12 members. Five girls play on both teams. How many girls will be at a party held for both teams? Sol’n
This is an example of the principle of inclusion and exclusion for two sets.
In general, for two sets A and B:
When we consider three or more sets, the expressions become more difficult. The key is a good Venn diagram. EX4. In a certain high school, there are 100 “4U” students. There are 52 taking Advanced Functions, 38 taking Calculus and Vectors and 41 taking Data Management. Seventeen students take both C&V and DM, 22 take AF and C&V and 15 take DM and AF. Eight students take all three maths. a. How many students take 4U math? b. How many students take no math? c. How many students take only C&V? Sol’n Let A represent the set of students studying Advanced Functions Let C represent the set of students studying C&V. Let DM represent the set of students studying Data Management. Now use a Venn diagram! Start in the middle and work your way out.
HW. 1. For the sets S = the set of whole numbers less than 20; A = the set of numbers which are perfect squares; B = the set of positive even numbers. Describe or state: a. A’ b. B’ c. A c B d. A 1 B e. (A 1 B)’ f. (A c B)’ g. n(A’) h. n(B’) i. n(A c B) j. n(A 1 B) k. n((A 1 B)’) l. n((A c B)’) 2. Using the terms equal, disjoint, and subset, describe the relationship between the given sets. Draw a Venn diagram for each situation. a. Z is the set of integers b. T is the set of teachers at NWSS R is the set of real numbers M is the set of math teachers you have this year c. N is the set of natural numbers d. N is the set of countries in NATO P is the set of positive integers SA is the set of countries in South America 3. Consider each statement and determine if it is true or false for any sets A and B. Explain. a. A f A b. i f A c. AcS = A d. A1S = A e. n(A1B) # n(AcB) 4. Textbook, Pg 270 #1 parts i, ii, 2, 3, 4, 5, 6, 9 For #2, write your answers using set notation, if possible.
Answers 1. a. non-perfect squares b. zero, plus all odd numbers c. {0, 1, 2, 4, 6, 8, 9, 10, 12, 14, 16, 18} e. all numbers from 0 to 19, except 4, 16 f. {3, 5, 7, 11, 13, 15, 17, 19} g. 15 h. 11 k. 18 l. 8 2. a. Z f R b. M f T c. N = P d. SA, N are disjoint 3. a. T b. T c. F, AcS = S d. T e. T d. {4, 16} i. 12 j. 2