DP SL Studies Chapter 7 Sets and Venn Diagrams

DP Studies Chapter 7 Homework Section A: 1, 2, 4, 5, 7, 9 Section B: 2, 4 Section C: 1, 2, 4, 5 Section D: 1, 4, 5, 6 Section E: 1, 4 Section F: 1, 3, 4, 7, 8 Section G: 2, 6, 8

Contents: Sets and Venn diagrams A Sets B Set builder notation C Complements of sets D Venn diagrams E Venn diagram regions F Numbers in regions G Problem solving with Venn diagrams

A. Set Notations A set is a collection of numbers or objects. Examples: 1. the set of all digits which we use to write numbers is {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. 2. set of all vowels, then V = {a, e, i, o, u}.

A. Set Notations The numbers or objects in a set are called the elements or members of the set. Examples: 1. So, for the set A = {1, 2, 3, 4, 5, 6, 7} we can say 4  A (4 is an element of set A), but 9  A (9 is not an element of set A). 2. For the set of all vowels, V = {a, e, i, o, u}, we can say a  V (a is an element of set V), but b  V (b is not an element of set V).

A. Set Notations The set { } or 0 is called the empty set and contains no elements. Example Let A be the set of all NBA players who are 10 feet tall. A = {}

A. Set Notations Special number sets

A. Set Notations The number of elements in set A is written n(A). Example: the set A = {2, 3, 5, 8, 13, 21} has 6 elements, so we write n(A) = 6.

A. Set Notations A set which has a finite number of elements is called a finite set. Example: 1. A = {2, 3, 5, 8, 13, 21} is a finite set. 2. Ø is also a finite set, since n(Ø) = 0.

A. Set Notations Infinite sets are sets which have infinitely many elements. Example: 1. the set of positive integers {1, 2, 3, 4,....} does not have a largest element, but rather keeps on going forever. It is therefore an infinite set. 2. the sets N, Z, Z +, Z –, Q, and R are all infinite sets.

A. Set Notations Suppose P and Q are two sets. P is a subset of Q if every element of P is also an element of Q. We write P Q. Example: {2, 3, 5} {1, 2, 3, 4, 5, 6} as every element in the first set is also in the second set. We say P is a proper subset of Q if P is a subset of Q but is not equal to Q. We write P Q.

A. Set Notations

Two sets are disjoint or mutually exclusive if they have no elements in common. Example: Set A = {0, 2, 4, 6, 8} and Set B = {1, 3, 5, 7} Set A and Set B are disjoint or mutually exclusive

A. Set Notations Example 1:

A. Set Notations Solution to Example 1:

B: Set Builder Notation Reading a set notation: A = {x | -2 < x < 4, x  Z} “the set of all x such that x is an integer between -2 and 4, including -2 and 4.” We can represent A on a number line as: A is a finite set, and n(A) = 7. such thatthe set of all

B: Set Builder Notation Reading a set notation: B = {x | -2 < x < 4, x  R} “the set of all real x such that x is greater than or equal to -2 and less than 4.” We represent B on a number line as: B is an infinite set, and n(B) = ∞

B: Set Builder Notation Example 2:

Solution to example 2:

C. Complement s of sets The symbol U is used to represent the universal set under consideration. Example: Suppose we are only interested in the natural numbers from 1 to 20, and we want to consider subsets of this set. We say the set U = {x | 1 < x < 20, x  N } is the universal set in this situation.

C. Complement s of sets The complement of A, denoted A’, is the set of all elements of U which are not in A. Example: If the universal set U = {1, 2, 3, 4, 5, 6, 7, 8}, and the set A = {1, 3, 5, 7, 8}, then the complement of A is A’ = {2, 4, 6}.

C. Complement s of sets

Example 3:

C. Complement s of sets Solution to example 3:

C. Complement s of sets Example 4:

C. Complement s of sets Solution to example 4:

C. Complement s of sets Example 5:

C. Complement s of sets Solution to example 5:

D. Venn Diagrams Venn diagrams are often used to represent sets of objects, numbers, or things. A Venn diagram consists of a universal set U represented by a rectangle. Sets within the universal set are usually represented by circles. Example:

D. Venn Diagrams Example of a universe set, U = {2, 3, 5, 7, 8}, A = {2, 7, 8}, and A’ = {3, 5}.

D. Venn Diagrams SUBSETS If B A then every element of B is also in A. The circle representing B is placed within the circle representing A.

D. Venn Diagrams

Example 6:

Solution to example 6:

Example 7:

Solution to example 7:

E. Venn Diagram Region The shading representations of Venn Diagrams.

Example 8:

Solution to example 8:

F. Numbers in Regions The four regions of the Venn Diagram that contains two overlapping of sets A and B.

F. Numbers in Regions Example 9:

F. Numbers in Regions Solution to Example 9:

F. Numbers in Regions

Solution to example 10:

G. Problem solving with Venn Diagrams Example 11: A squash club has 27 members. 19 have black hair, 14 have brown eyes, and 11 have both black hair and brown eyes. a. Place this information on a Venn diagram. b. Hence find the number of members with: i. black hair or brown eyes ii. black hair, but not brown eyes.

G. Problem solving with Venn Diagrams Solution to example 11:

G. Problem solving with Venn Diagrams Example 12: A platform diving squad of 25 has 18 members who dive from 10 m and 17 who dive from 4 m. How many dive from both platforms?

G. Problem solving with Venn Diagrams Solution to example 12:

G. Problem solving with Venn Diagrams Example 13: A city has three football teams in the national league: A, B, and C. In the last season, 20% of the city’s population saw team A play, 24% saw team B, and 28% saw team C. Of these, 4% saw both A and B, 5% saw both A and C, and 6% saw both B and C. 1% saw all three teams play. Using a Venn diagram, find the percentage of the city’s population which: a. saw only team A play b. saw team A or team B play but not team C c. did not see any of the teams play.

Solution to example 13: